American Institute of Mathematical Sciences

February  2014, 8(1): 173-197. doi: 10.3934/ipi.2014.8.173

Geometric reconstruction in bioluminescence tomography

 1 Fakultät für Mathematik, Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie (KIT), D-76128 Karlsruhe, Germany, Germany

Received  March 2012 Revised  July 2013 Published  March 2014

In bioluminescence tomography the location as well as the radiation intensity of a photon source (marked cell clusters) inside an organism have to be determined given the outside photon count. This inverse source problem is ill-posed: it suffers not only from strong instability but also from non-uniqueness. To cope with these difficulties the source is modeled as a linear combination of indicator functions of measurable domains leading to a nonlinear operator equation. The solution process is stabilized by a Tikhonov like functional which penalizes the perimeter of the domains. For the resulting minimization problem existence of a minimizer, stability, and regularization property are shown. Moreover, an approximate variational principle is developed based on the calculated domain derivatives which states that there exist smooth almost stationary points of the Tikhonov like functional near to any of its minimizers. This is a crucial property from a numerical point of view as it allows to approximate the searched-for domain by smooth domains. Based on the theoretical findings numerical schemes are proposed and tested for star-shaped sources in 2D: computational experiments illustrate performance and limitations of the considered approach.
Citation: Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems and Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173
References:
 [1] K. Atkinson and W. Han, Theoretical Numerical Analysis, 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4. [2] H. Attouch, G. Buttazzo and G.Michaille, Variational Analysis in Sobolev and BV Space, MPS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. [3] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. doi: 10.1088/0266-5611/25/5/053001. [4] M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988. [5] M. Burger and S. J. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182. [6] F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123. [7] W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner and A. Cong, Practical reconstruction method for bioluminescence tomography, Opt. Express, 13 (2005), 6756-6771. doi: 10.1364/OPEX.13.006756. [8] C. H. Contag and B. D. Ross, It's not just about anatomy: In vivo bioluminescence imaging as an eyepiece into biology, Journal of Magnetic Resonance Imaging, 16 (2002), 378-387. doi: 10.1002/jmri.10178. [9] A. De Cezaro and A. Leitão, Level-set approaches of $L_2$-type for recovering shape and contrast in ill-posed problems, Inverse Probl. Sci. Eng., 20 (2012), 571-587. doi: 10.1080/17415977.2011.639452. [10] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719826. [11] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [12] ______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6. [13] W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1998. [14] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser, Boston, 1984. [15] W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008. [16] M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens, 3rd edition, Vieweg + Teubner, Wiesbaden, 2009. doi: 10.1007/978-3-8348-9309-3. [17] H. Harbrecht and J. Tausch, An efficient numerical method for a shape-identification problem arising from the heat equation, Inverse Problems, 27 (2011), 065013, 18 pp. doi: 10.1088/0266-5611/27/6/065013. [18] F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, Springer, 2009. doi: 10.1007/978-1-4020-8839-1_3. [20] C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611970920. [21] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006. [22] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. [23] _______, Regularization of ill-posed Mumford-Shah models with perimeter penalization, Inverse Problems, 26 (2010), 115001, 25 pp. doi: 10.1088/0266-5611/26/11/115001. [24] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687. doi: 10.1080/01630563.1980.10120631. [25] G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299. doi: 10.1118/1.1766420. [26] R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128. doi: 10.1038/nm0103-123. [27] H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472. doi: 10.2307/2371513.

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References:
 [1] K. Atkinson and W. Han, Theoretical Numerical Analysis, 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4. [2] H. Attouch, G. Buttazzo and G.Michaille, Variational Analysis in Sobolev and BV Space, MPS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. [3] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. doi: 10.1088/0266-5611/25/5/053001. [4] M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988. [5] M. Burger and S. J. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182. [6] F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123. [7] W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner and A. Cong, Practical reconstruction method for bioluminescence tomography, Opt. Express, 13 (2005), 6756-6771. doi: 10.1364/OPEX.13.006756. [8] C. H. Contag and B. D. Ross, It's not just about anatomy: In vivo bioluminescence imaging as an eyepiece into biology, Journal of Magnetic Resonance Imaging, 16 (2002), 378-387. doi: 10.1002/jmri.10178. [9] A. De Cezaro and A. Leitão, Level-set approaches of $L_2$-type for recovering shape and contrast in ill-posed problems, Inverse Probl. Sci. Eng., 20 (2012), 571-587. doi: 10.1080/17415977.2011.639452. [10] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719826. [11] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [12] ______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6. [13] W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1998. [14] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser, Boston, 1984. [15] W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008. [16] M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens, 3rd edition, Vieweg + Teubner, Wiesbaden, 2009. doi: 10.1007/978-3-8348-9309-3. [17] H. Harbrecht and J. Tausch, An efficient numerical method for a shape-identification problem arising from the heat equation, Inverse Problems, 27 (2011), 065013, 18 pp. doi: 10.1088/0266-5611/27/6/065013. [18] F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, Springer, 2009. doi: 10.1007/978-1-4020-8839-1_3. [20] C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611970920. [21] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006. [22] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. [23] _______, Regularization of ill-posed Mumford-Shah models with perimeter penalization, Inverse Problems, 26 (2010), 115001, 25 pp. doi: 10.1088/0266-5611/26/11/115001. [24] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687. doi: 10.1080/01630563.1980.10120631. [25] G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299. doi: 10.1118/1.1766420. [26] R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128. doi: 10.1038/nm0103-123. [27] H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472. doi: 10.2307/2371513.
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