American Institute of Mathematical Sciences

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 and 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), 076002. doi: 10.1117/1.JBO.19.7.076002. [2] W. Cong, F. Liu, C. Wang and G. Wang, X-ray micro-modulated luminescence tomography (XMLT), Opt. Express, 22 (2014), 5572-5580. doi: 10.1364/OE.22.005572. [3] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. [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), 89-108. doi: 10.1007/s12220-007-9007-6. [5] R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations, Med. Phys., 26 (1999), 1832-1837. doi: 10.1118/1.598688. [6] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353. [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), 1554p. [8] V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology, Nat. Methods, 7 (2010), 603-614. doi: 10.1038/nmeth.1483. [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), 313-320. doi: 10.1038/nbt1074. [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), 43-71. doi: 10.1145/355984.355989. [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), 1992-1999. doi: 10.1109/TMI.2010.2055883. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Scattering Theory, Academic Press, New York, 1979. [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), 1779-1792. doi: 10.1118/1.597634. [14] P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017. [15] ________, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp. doi: 10.1088/0266-5611/25/7/075011. [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), 7801-7809. doi: 10.1364/OE.14.007801. [17] L. V. Wang and S. Hu, Photoacoustic tomography: In vivo imaging from organelles to organs, Science, 335 (2012), 1458-1462. doi: 10.1126/science.1216210. [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.

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), 076002. doi: 10.1117/1.JBO.19.7.076002. [2] W. Cong, F. Liu, C. Wang and G. Wang, X-ray micro-modulated luminescence tomography (XMLT), Opt. Express, 22 (2014), 5572-5580. doi: 10.1364/OE.22.005572. [3] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. [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), 89-108. doi: 10.1007/s12220-007-9007-6. [5] R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations, Med. Phys., 26 (1999), 1832-1837. doi: 10.1118/1.598688. [6] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353. [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), 1554p. [8] V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology, Nat. Methods, 7 (2010), 603-614. doi: 10.1038/nmeth.1483. [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), 313-320. doi: 10.1038/nbt1074. [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), 43-71. doi: 10.1145/355984.355989. [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), 1992-1999. doi: 10.1109/TMI.2010.2055883. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Scattering Theory, Academic Press, New York, 1979. [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), 1779-1792. doi: 10.1118/1.597634. [14] P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017. [15] ________, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp. doi: 10.1088/0266-5611/25/7/075011. [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), 7801-7809. doi: 10.1364/OE.14.007801. [17] L. V. Wang and S. Hu, Photoacoustic tomography: In vivo imaging from organelles to organs, Science, 335 (2012), 1458-1462. doi: 10.1126/science.1216210. [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.
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