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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