American Institute of Mathematical Sciences

Advanced
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, (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, (2006). Google Scholar

[3]

G. Bal, Inverse transport theory and applications,, Inverse Problems, 25 (2009). 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, (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. 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. 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. 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. 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. 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, (2001). doi: 10.1137/1.9780898719826. Google Scholar

[11]

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

[12]

______, Nonconvex minization problems,, Bull. Am. Math. Soc., 1 (1979), 443. 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, (1998). Google Scholar

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, (1984). Google Scholar

[15]

W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659. doi: 10.1088/0266-5611/22/5/008. Google Scholar

[16]

M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens,, 3rd edition, (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). doi: 10.1088/0266-5611/27/6/065013. Google Scholar

[18]

F. Hettlich, The Domain Derivative in Inverse Obstacle Problems,, Habilitation thesis, (1999). Google Scholar

[19]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Mathematical Modelling: Theory and Applications, (2009). doi: 10.1007/978-1-4020-8839-1_3. Google Scholar

[20]

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

[21]

J. Nocedal and S. J. Wright, Numerical Optimization,, 2nd edition, (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. 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). 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. 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), 31 (2004), 2289. doi: 10.1118/1.1766420. Google Scholar

[26]

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

[27]

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

show all references

References:
[1]

K. Atkinson and W. Han, Theoretical Numerical Analysis,, 3rd edition, (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, (2006). Google Scholar

[3]

G. Bal, Inverse transport theory and applications,, Inverse Problems, 25 (2009). 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, (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. 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. 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. 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. 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. 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, (2001). doi: 10.1137/1.9780898719826. Google Scholar

[11]

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

[12]

______, Nonconvex minization problems,, Bull. Am. Math. Soc., 1 (1979), 443. 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, (1998). Google Scholar

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, (1984). Google Scholar

[15]

W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659. doi: 10.1088/0266-5611/22/5/008. Google Scholar

[16]

M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens,, 3rd edition, (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). doi: 10.1088/0266-5611/27/6/065013. Google Scholar

[18]

F. Hettlich, The Domain Derivative in Inverse Obstacle Problems,, Habilitation thesis, (1999). Google Scholar

[19]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Mathematical Modelling: Theory and Applications, (2009). doi: 10.1007/978-1-4020-8839-1_3. Google Scholar

[20]

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

[21]

J. Nocedal and S. J. Wright, Numerical Optimization,, 2nd edition, (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. 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). 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. 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), 31 (2004), 2289. doi: 10.1118/1.1766420. Google Scholar

[26]

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

[27]

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

[1]

Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations & Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011
[2]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825
[3]

Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems & Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012
[4]

Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008
[5]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[6]

Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397
[7]

Plamen Stefanov, Gunther Uhlmann. Instability of the linearized problem in multiwave tomography of recovery both the source and the speed. Inverse Problems & Imaging, 2013, 7 (4) : 1367-1377. doi: 10.3934/ipi.2013.7.1367
[8]

Shui-Nee Chow, Ke Yin, Hao-Min Zhou, Ali Behrooz. Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 79-102. doi: 10.3934/ipi.2014.8.79
[9]

Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems & Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291
[10]

Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069
[11]

Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems & Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051
[12]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[13]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
[14]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[15]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
[16]

Biao Qu, Naihua Xiu. A relaxed extragradient-like method for a class of constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 645-654. doi: 10.3934/jimo.2007.3.645
[17]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
[18]

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems & Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1
[19]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[20]

Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences