扩散方程在生物体内成像的应用研究
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摘要
分子影像学是一门新兴的交叉学科,涉及到分子生物学、物理学、数学、信息科学、放射医学等多个学科。分子成像理论是分子影像学的重要研究内容。自发荧光断层成像(BLT)是一种生物医学分子成像技术,它通过对体内的光源进行局部化和量化来揭示细胞及分子的活动信息。扩散方程及Robin边界条件能够很好地描述光子在生物组织中的传播,是BLT问题中的一个热点研究课题。本文分别将重叠区域算法,非重叠区域算法,混合有限元法,差分法应用于扩散方程的求解,推导出求解扩散方程的离散化格式,并建立重构光源的数学模型。进行了重叠区域算法的仿真实验,通过利用求解BLT的正向问题获得仿体表面光源能量分布,然后应用遗传算法对光源进行重建。对于所给区域我们采用四面体网格化,通过确定光源可行域,提高解的稳定性和有效性。最后应用MOSE算法平台在3D和real环境下验证MC算法的的稳定性并与解析解进行了比较。
Molecular imaging is a new interdisciplinary subject, involving the molecular biology, physics, mathematics, the information science, the radiation medicine and so on. The emphasis research content is the molecular imaging theory. Bioluminescence Tomography (BLT) allows in vivo localization and quantification of bioluminescent sources inside a small animal to reveal various molecular and cellular activities. Photon propagation in biological tissues can be well described by a steady-state diffusion equation and Robin-type boundary conditions. In this paper, the overlap domain decomposition method (ODDM), non-overlap domain decomposition method, mixed finite element method and finite difference method are proposed respectively. Some kinds of discretization of the diffusion equation are obtained. The mathematic model of the inverse problem is constructed to determine the position of the light source. The simulation result of the ODDM is given. We can obtain the photon density distribution of the object surface, as well as reconstruct the position of the light source by using ODDM and genetic algorithm. The experimental region is partitioned into a certain number sub-regions. The stability and efficiency of the solution are enhanced by predefining the permissible region. Finally, we confirm the stability of MC algorithm in 3D and the real environment, comparing with the analytic solution by using MOSE platform.
Molecular imaging is a new interdisciplinary subject, involving the molecular biology, physics, mathematics, the information science, the radiation medicine and so on. The emphasis research content is the molecular imaging theory. Bioluminescence Tomography (BLT) allows in vivo localization and quantification of bioluminescent sources inside a small animal to reveal various molecular and cellular activities. Photon propagation in biological tissues can be well described by a steady-state diffusion equation and Robin-type boundary conditions. In this paper, the overlap domain decomposition method (ODDM), non-overlap domain decomposition method, mixed finite element method and finite difference method are proposed respectively. Some kinds of discretization of the diffusion equation are obtained. The mathematic model of the inverse problem is constructed to determine the position of the light source. The simulation result of the ODDM is given. We can obtain the photon density distribution of the object surface, as well as reconstruct the position of the light source by using ODDM and genetic algorithm. The experimental region is partitioned into a certain number sub-regions. The stability and efficiency of the solution are enhanced by predefining the permissible region. Finally, we confirm the stability of MC algorithm in 3D and the real environment, comparing with the analytic solution by using MOSE platform.
引文
[1] Ralph Weissleder. Molecular imaging: Exploring the next frontier. Radiology. 1999, 212: 609-614.
[2] V. Ntziachristos, J. Ripoll, L. V. Wang, R. Weissleder. “Looking and listening to light: the evolution of whole-body photonic imaging”. Nature Biotechnology. 2005, 3(3), 313-320,
[3] R. W. Rice, M. D. Cable and M. B. Nelson. “In vivo imaging of light-emitting probes”. Journal of Biomedical Optics. 6(4), 2001, 432-440.
[4] Tarik F. Massoud and Sanjiv S. Gambhir. “Molecular imaging in living subjects: seeing fundamental biological processes in a new light”. Genes and development. 2003, 17, 545-580.
[5] G. Wang, Y. Li, and M. Jiang. “Uniqueness theorems in bioluminescence tomography”. Med, Phys. 2004, 31, 2289-2299.
[6] G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter and J. F. Meinel, “Development of the first bioluminescent CT scanner”. Radiology. 2003, 229 (P): 566.
[7] H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method”. Acad. Radiol. 2004, 11, 1029-1038.
[8] W. Cong, D. Kumar, Y. Liu, A. Cong and G. Wang. “A practical method to determine the light source distribution in bioluminescent imaging”. Proc.SPIE. 2004, 5535, 679-686.
[9] Y. J. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W, Yang and H. Li. “A multilevel adaptive finite element algorithm for bioluminescence tomography”. OPTICS EXPRESS. 2006, 14, 8211-8223.
[10] W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P.B. McCray, J. Zabner, and A.Cong. “Practical reconstruction methods for bioluminescence tomography”. OPTICS EXPRESS. 2005, 13, 6756-6771.
[11] M. Jiang and G. Wang. “Image reconstruction for bioluminescence tomography”. Proc. SPIE. 2004, 5535, 335-351.
[12] 李慧,田捷,王革. 基于 Monte Carlo 在体生物光学成像光子传输模型.软件 学报,2004,15(11): 1709-1719.
[13] M. Dryja and O. B. Widlund. “Towards a unified theory of domaindecomposition algorithms for elliptic problems”. Iterative methods for large systems, Academic press, 1990.
[14] R. Nabben and C. Vuik. “Domain Decomposition methods and deflated krylov subspace iterations,”“European conference on computational fluid dynamics” . 2006
[15] Q. Du and D. Yu. “A domain decomposition method based on natural boundary reduction for nonlinear time-dependent exterior wave problems”. Neural Computing and Applications. 2002, 68, 111-129.
[16] T. F Chan and T. P. Mathew. “Domain decomposition algorithms”. Acta Numerica. 1994, pp 61-143
[17] J. Xu. “Iterative methods by space decomposition and subspace correction”. SIAM Rev. 1992, 34, 581-613.
[18] X. C. Tai. “Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities”. Numer. Math.. 2003, 93, 755-86.
[19] M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy. “The finite element approach for the propagation of light in scattering media: Boundary and source condition”. Med. Phys. 1995, 22, 1779-1792.
[20] Wenxiang Cong, Lihong V Wang and Ge Wang, “formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous meiuum” . “BioMedical Engineering OnLine”. 2004.
[21] G. Wang, M. Jiang, J. Tian,W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, T. Zhou, J. Cheng, Y. Lv, H. Li and J. Luo, “Recent development in bioluminescence tomography”. presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA, 6-9 Apr. (2006).
[22] R. Nabben, “Comparisons between multiplicative and additive Schwarz iterations in domain decomposition methods”. Num. Math.. 95, 145-162 (2003).
[23] W. Fang, T. Wu and P. Chen, “An algorithm Global Optimization for Rational Functions with Rational Constraints”. Journal of Global Optimization. 2000, 18, 211-218.
[24] M. H. Lim, Y. Yuan and S. Omatu, “Efficient genetic algorithms using simple genes exchange local search policy for the quadratic Assignment problem”. Computational optimization and Applications. 2000, 15, 249-268.
[25] S. Chattopadhyay and N. Choudhary, “Genetic Algorithm based Approach for Low Power combinational circuit testing”. “Proceedings of the 16th international conference on VLSI Design (VLS’ 03),” IEEE. (2003).
[26] A. Frommer, R. Nabben., D. B. Szyld, “Weighted max norms, splittings, andoverlapping additive Schwarz iterations”. Numer. Math. 1999, 83, 259-278.
[27] A. Frommer, D. B. Szyld, , “An algebraic convergence theory for restricted additive Schwarz methods using weighted max norm”. SIAM Journal on Numerical Analysis. 2001, 39, 463-479.
[28] Ivo Marec and Danilel B. Szyld. “Algebraic analysis of Schwarz Methods for singular systems”. Lecture Note Computational Science and Engineering. 2004, 40, 647-652.
[29] R. A. Horm and C. R. Johnson, “Matrix Analysis”. Post & Telecom Press.1986, 257-335.
[30] Seongjai Kin, “Domain decomposition iterative procedures for solving scalar waves in frequency domain” Numer.Math.(1998)79:231-259
[31] Lions, P. L.. “On the Schwarz alternating methods: a variant for nonoverlapping sub-domains”. in Third International Symposium on Domain Dccomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux and O. B. Wildlund, eds., SIAM, Philadelphia, PA, 1990, pp. 202-231
[32] de La Bourdonnaye, Farhat, Macedo, Mogoules, Roux. “A non overlapping demain decomposition method for the exterior Hellmholtz problem”. INRIA. 1997, 32pages
[33] Magne S. Espedal, Xue-Cheng Tai and Ningning Yan, “A hybrid non-overlapping domain decomposition scheme for advection dominated advection-diffusion problems” Numerical Algorithms.1998. 18, 321-336.
[34] G. Lube, L. Muller and F. C. Otto. “A non-overlapping Domain decomposition method for the advection-diffusion problem”. Computing. 2000, 64, 49-68.
[35] Doninik Shchotzau, “Mixed finite element methods for stationary incompressible magneto-hydrodynamics”. Numer. Math. 2004, 96: 771-800.
[36] 徐长发.实用偏微分方程数值解法. 武汉:华中理工大学出版.2000.
[37] 胡健伟,汤怀民.偏微分方程数值方法.第二版.北京:科学出版社,1999.
[38] 陆金甫.偏微分方程数值解. 北京:清华大学出版社.2004.
[39] 罗振东.混合有限元算法基础及其应用.北京:科学出版社,2006.
[40] F. Brezzi, M. Fortin, “A minimal stabilization procedure for mixed finite element methods”, Numer. Math. 2001, 89: 457-491.
[41] Eckhard Schneid, Peter Knabner, Florin Radu, “A priori error estimates for a mixed finite element discretization for Richards’ equation”. Numer. Math. 2004, 98: 353-370.
[42] L. Guo, H. Chen, “ H1-Galerkon mixed finite element method for the regularized long wave equation”. Computing, 2006, 77, 205-221.
[43] R. L. Talor, F. C. Filippou, A, Saritas, F. Auricchio, “A mixied finite elementmethod for beam and frame problems”. Computing Mechanics. 2003, 31,129-203.
[44] Wang LH, Jacques SL. Mnote Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C. M. D. Anderson Cancer Center, University of Texas, 1992.7-10.
[2] V. Ntziachristos, J. Ripoll, L. V. Wang, R. Weissleder. “Looking and listening to light: the evolution of whole-body photonic imaging”. Nature Biotechnology. 2005, 3(3), 313-320,
[3] R. W. Rice, M. D. Cable and M. B. Nelson. “In vivo imaging of light-emitting probes”. Journal of Biomedical Optics. 6(4), 2001, 432-440.
[4] Tarik F. Massoud and Sanjiv S. Gambhir. “Molecular imaging in living subjects: seeing fundamental biological processes in a new light”. Genes and development. 2003, 17, 545-580.
[5] G. Wang, Y. Li, and M. Jiang. “Uniqueness theorems in bioluminescence tomography”. Med, Phys. 2004, 31, 2289-2299.
[6] G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter and J. F. Meinel, “Development of the first bioluminescent CT scanner”. Radiology. 2003, 229 (P): 566.
[7] H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method”. Acad. Radiol. 2004, 11, 1029-1038.
[8] W. Cong, D. Kumar, Y. Liu, A. Cong and G. Wang. “A practical method to determine the light source distribution in bioluminescent imaging”. Proc.SPIE. 2004, 5535, 679-686.
[9] Y. J. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W, Yang and H. Li. “A multilevel adaptive finite element algorithm for bioluminescence tomography”. OPTICS EXPRESS. 2006, 14, 8211-8223.
[10] W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P.B. McCray, J. Zabner, and A.Cong. “Practical reconstruction methods for bioluminescence tomography”. OPTICS EXPRESS. 2005, 13, 6756-6771.
[11] M. Jiang and G. Wang. “Image reconstruction for bioluminescence tomography”. Proc. SPIE. 2004, 5535, 335-351.
[12] 李慧,田捷,王革. 基于 Monte Carlo 在体生物光学成像光子传输模型.软件 学报,2004,15(11): 1709-1719.
[13] M. Dryja and O. B. Widlund. “Towards a unified theory of domaindecomposition algorithms for elliptic problems”. Iterative methods for large systems, Academic press, 1990.
[14] R. Nabben and C. Vuik. “Domain Decomposition methods and deflated krylov subspace iterations,”“European conference on computational fluid dynamics” . 2006
[15] Q. Du and D. Yu. “A domain decomposition method based on natural boundary reduction for nonlinear time-dependent exterior wave problems”. Neural Computing and Applications. 2002, 68, 111-129.
[16] T. F Chan and T. P. Mathew. “Domain decomposition algorithms”. Acta Numerica. 1994, pp 61-143
[17] J. Xu. “Iterative methods by space decomposition and subspace correction”. SIAM Rev. 1992, 34, 581-613.
[18] X. C. Tai. “Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities”. Numer. Math.. 2003, 93, 755-86.
[19] M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy. “The finite element approach for the propagation of light in scattering media: Boundary and source condition”. Med. Phys. 1995, 22, 1779-1792.
[20] Wenxiang Cong, Lihong V Wang and Ge Wang, “formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous meiuum” . “BioMedical Engineering OnLine”. 2004.
[21] G. Wang, M. Jiang, J. Tian,W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, T. Zhou, J. Cheng, Y. Lv, H. Li and J. Luo, “Recent development in bioluminescence tomography”. presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA, 6-9 Apr. (2006).
[22] R. Nabben, “Comparisons between multiplicative and additive Schwarz iterations in domain decomposition methods”. Num. Math.. 95, 145-162 (2003).
[23] W. Fang, T. Wu and P. Chen, “An algorithm Global Optimization for Rational Functions with Rational Constraints”. Journal of Global Optimization. 2000, 18, 211-218.
[24] M. H. Lim, Y. Yuan and S. Omatu, “Efficient genetic algorithms using simple genes exchange local search policy for the quadratic Assignment problem”. Computational optimization and Applications. 2000, 15, 249-268.
[25] S. Chattopadhyay and N. Choudhary, “Genetic Algorithm based Approach for Low Power combinational circuit testing”. “Proceedings of the 16th international conference on VLSI Design (VLS’ 03),” IEEE. (2003).
[26] A. Frommer, R. Nabben., D. B. Szyld, “Weighted max norms, splittings, andoverlapping additive Schwarz iterations”. Numer. Math. 1999, 83, 259-278.
[27] A. Frommer, D. B. Szyld, , “An algebraic convergence theory for restricted additive Schwarz methods using weighted max norm”. SIAM Journal on Numerical Analysis. 2001, 39, 463-479.
[28] Ivo Marec and Danilel B. Szyld. “Algebraic analysis of Schwarz Methods for singular systems”. Lecture Note Computational Science and Engineering. 2004, 40, 647-652.
[29] R. A. Horm and C. R. Johnson, “Matrix Analysis”. Post & Telecom Press.1986, 257-335.
[30] Seongjai Kin, “Domain decomposition iterative procedures for solving scalar waves in frequency domain” Numer.Math.(1998)79:231-259
[31] Lions, P. L.. “On the Schwarz alternating methods: a variant for nonoverlapping sub-domains”. in Third International Symposium on Domain Dccomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux and O. B. Wildlund, eds., SIAM, Philadelphia, PA, 1990, pp. 202-231
[32] de La Bourdonnaye, Farhat, Macedo, Mogoules, Roux. “A non overlapping demain decomposition method for the exterior Hellmholtz problem”. INRIA. 1997, 32pages
[33] Magne S. Espedal, Xue-Cheng Tai and Ningning Yan, “A hybrid non-overlapping domain decomposition scheme for advection dominated advection-diffusion problems” Numerical Algorithms.1998. 18, 321-336.
[34] G. Lube, L. Muller and F. C. Otto. “A non-overlapping Domain decomposition method for the advection-diffusion problem”. Computing. 2000, 64, 49-68.
[35] Doninik Shchotzau, “Mixed finite element methods for stationary incompressible magneto-hydrodynamics”. Numer. Math. 2004, 96: 771-800.
[36] 徐长发.实用偏微分方程数值解法. 武汉:华中理工大学出版.2000.
[37] 胡健伟,汤怀民.偏微分方程数值方法.第二版.北京:科学出版社,1999.
[38] 陆金甫.偏微分方程数值解. 北京:清华大学出版社.2004.
[39] 罗振东.混合有限元算法基础及其应用.北京:科学出版社,2006.
[40] F. Brezzi, M. Fortin, “A minimal stabilization procedure for mixed finite element methods”, Numer. Math. 2001, 89: 457-491.
[41] Eckhard Schneid, Peter Knabner, Florin Radu, “A priori error estimates for a mixed finite element discretization for Richards’ equation”. Numer. Math. 2004, 98: 353-370.
[42] L. Guo, H. Chen, “ H1-Galerkon mixed finite element method for the regularized long wave equation”. Computing, 2006, 77, 205-221.
[43] R. L. Talor, F. C. Filippou, A, Saritas, F. Auricchio, “A mixied finite elementmethod for beam and frame problems”. Computing Mechanics. 2003, 31,129-203.
[44] Wang LH, Jacques SL. Mnote Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C. M. D. Anderson Cancer Center, University of Texas, 1992.7-10.