American Institute of Mathematical Sciences

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

Geometric reconstruction in bioluminescence tomography

Tim Kreutzmann 1, and  Andreas Rieder 1,

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.
Keywords: Bioluminescence tomography, shape optimization., domain derivative, inverse source problem, Tikhonov like regularization.
Mathematics Subject Classification: Primary: 65J15, 65J2.
Citation: Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems & 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.  Google Scholar

[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. Google Scholar

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G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar

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M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

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F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. Google Scholar

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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.  Google Scholar

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C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611970920.  Google Scholar

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J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299. doi: 10.1118/1.1766420.  Google Scholar

[26]

R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128. doi: 10.1038/nm0103-123.  Google Scholar

[27]

H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472. doi: 10.2307/2371513.  Google Scholar

show all references

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.  Google Scholar

[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. Google Scholar

[3]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar

[4]

M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[12]

______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

[13]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1998.  Google Scholar

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser, Boston, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. Google Scholar

[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.  Google Scholar

[20]

C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611970920.  Google Scholar

[21]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299. doi: 10.1118/1.1766420.  Google Scholar

[26]

R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128. doi: 10.1038/nm0103-123.  Google Scholar

[27]

H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472. doi: 10.2307/2371513.  Google Scholar

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