Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T10:27:45.719Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Coupled Complex Boundary Method for Bioluminescence Tomography

Part of: Partial differential equations, boundary value problems Existence theories Numerical methods Numerical linear algebra

Published online by Cambridge University Press:  15 January 2016

Rongfang Gong ,
Xiaoliang Cheng  and
Weimin Han
Rongfang Gong*
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Xiaoliang Cheng
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Weimin Han
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
*
*Corresponding author. Email addresses:grf_math@nuaa.edu.cn (R. Gong), xiaoliangcheng@zju.edu.cn (X. Cheng), weimin-han@uiowa.edu (W. Han)
Get access

Abstract

In this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

Keywords

Bioluminescence tomographyTikhonov regularizationconvergence ratefinite element methodserror estimate

MSC classification

Secondary: 65N21: Inverse problems 65F22: Ill-posedness, regularization 49J40: Variational methods including variational inequalities 74S05: Finite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 226 - 250
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Atkinson, K. and Han, W., Theoretical Numerical Analysis: A Functional Analysis Framework, Third edition, Springer-Verlag, New York, 2009.Google Scholar
[2]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Third edition, Springer-Verlag, New York, 2008.Google Scholar
[3]Cheng, X. L., Gong, R. F. and Han, W., A new general mathematical framework for bioluminescence tomography, Comput. Methods Appl. Mech. Engrg., 197 (2008), 524–535.Google Scholar
[4]Cheng, X. L., Gong, R. F., Han, W. and Zheng, X., A novel coupled complex boundary method for solving inverse source problems, Inverse Probl., 30 (2014), 055002 20pp.Google Scholar
[5]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[6]Cong, W. X., Wang, G., Kumar, D.et al., Practical reconstrucion method for bioluminescence tomography, Opt. Express, 18 (2005), 6756–6771.Google Scholar
[7]Dautray, R. and Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer, Berlin, 1988.Google Scholar
[8]Feng, J., Qin, C., Zhu, S.et al., Total variation regularization for bioluminescence tomography with the split Bregman method, Appl. Opt., 51 (2012), 4501–4512.Google Scholar
[9]Gong, R. F., Cheng, X. L. and Han, W., Theorectical analysis and numerical realization of bioluminesecne tomography, J. Concrete Appl. Math., 8 (2010), 504–527.Google Scholar
[10]Gong, R. F., Cheng, X. L. and Han, W., A fast solver for an inverse problem arising in bioluminesecne tomography, J. Comp. Appl. Math., 267 (2014) 228–243.Google Scholar
[11]Guo, W., Jia, K., Han, D.et al., Efficient sparse reconstruction algorithm for bioluminescence tomography based on duality and variable splitting, Appl. Opt., 51 (2002), 5676–5685.Google Scholar
[12]Han, W., Cong, W. X. and Wang, G., Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Probl., 22 (2006), 1659–1675.Google Scholar
[13]He, X. W., Hou, Y. B., Chen, D. F.et al., Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method, Int. J. Biomed. Imag. 2011 (2011), 20353711 pages.Google Scholar
[14]Isakov, V., Inverse Source Problems, American Mathematical Society, New York, 1990.Google Scholar
[15]Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996.Google Scholar
[16]Kreutzmann, T., Geometric Regularization in Bioluminescence Tomography, KIT Scientific Publishing, 2014.Google Scholar
[17]Liu, K., Tian, J., Lu, Y.et al., A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations, Opt. Express, 18 (2010), 3732–3745.Google Scholar
[18]Lv, Y., Tian, J., Cong, W.et al., A multilevel adaptive finite element algorithm for bioluminescence tomography, Opt. Express, 14 (2006), 8211–8223CrossRefGoogle ScholarPubMed
[19]Wang, G., Li, Y. and Jiang, M., Uniqueness theorems in bioluminescence tomography, Med. Phys., 31 (2004), 2289–2299.Google Scholar
[20]Wang, T., Gao, S., Zhang, L.et al., Overlap domain decomposition method for bioluminescence tomography (BLT), Commun. Numer. Meth. Engng., 26 (2010), 511–523.Google Scholar
[21]Weissleder, R. and Ntziachristos, V., Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123–128.Google Scholar