OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 10 — Oct. 31, 2007
« Show journal navigation

Image reconstruction for bioluminescence tomography from partial measurement

Ming Jiang, Tie Zhou, Jiantao Cheng, Wenxiang Cong, and Ge Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 18, pp. 11095-11116 (2007)
http://dx.doi.org/10.1364/OE.15.011095


View Full Text Article

Acrobat PDF (734 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The bioluminescence tomography is a novel molecular imaging technology for small animal studies. Known reconstruction methods require the completely measured data on the external surface, although only partially measured data is available in practice. In this work, we formulate a mathematical model for BLT from partial data and generalize our previous results on the solution uniqueness to the partial data case. Then we extend two of our reconstruction methods for BLT to this case. The first method is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Both methods allow the incorporation of knowledgebased constraints. Two practical constraints, the source non-negativity and support constraints, are introduced to regularize the BLT problem and produce stability. The initial choice of both methods and its influence on the regularization and stability are also discussed. The proposed algorithms are evaluated and validated with intensive numerical simulation and a physical phantom experiment. Quantitative results including the location and source power accuracy are reported. Various algorithmic issues are investigated, especially how to avoid the inverse crime in numerical simulations.

© 2007 Optical Society of America

1. Introduction

Gene therapy is a breakthrough in the modern medicine, which promises to cure diseases by modifying gene expressions. A key for the development of gene therapy is to monitor the in vivo gene transfer and its efficacy in the mouse model. Traditional biopsy methods are invasive, insensitive, inaccurate, inefficient, and limited in the extent. To map the distribution of the administered gene, reporter genes such as those producing luciferase are used to generate light signals within a living mouse. Bioluminescent imaging (BLI) is an optical technique for sensing gene expression, protein functions and other biological processes in mouse models by reporters such as luciferases that generate internal biological light sources [1

1. C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002). [CrossRef] [PubMed]

, 2

2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

]. The light emitted within the mouse can be captured externally using a highly sensitive CCD camera [3

3. B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001). [CrossRef] [PubMed]

]. “The collection times for BLI are relatively short compared to non-optical modalities, an advantage of optical imaging methods in general” [1

1. C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002). [CrossRef] [PubMed]

].

BLI has great potentials in various biomedical applications, including regenerative medicine, developmental therapeutics, treatment of residual minimal disease, and studies on cancer

stem cells. “Additionally, time-lapse whole body imaging of animals bearing xenografts of appropriately labeled cancer cells can provide new information on tumor growth dynamics and metastasis patterns that it is not possible to obtain by invasive experimental approaches” [4

4. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005). [CrossRef] [PubMed]

]. “Bioluminescence imaging (BLI) is a highly sensitive tool for visualizing tumors, neoplastic development, metastatic spread, and response to therapy”
[5

5. Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004). [CrossRef] [PubMed]

]. Although the spatial resolution is limited when compared with other imaging modalities, the advantages of BLI in vivo imaging are sensitivity, speed, non-invasiveness, cost, ease, and low background noise (in contrast to fluorescence, PET and other non-optical modalities) [1

1. C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002). [CrossRef] [PubMed]

]. BLI could be applied to study almost all diseases in small animal models [6

6. A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002). [CrossRef]

, 1

1. C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002). [CrossRef] [PubMed]

, 7

7. A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003). [CrossRef]

, 8

8. A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003). [PubMed]

, 9

9. J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003). [CrossRef] [PubMed]

, 10

10. C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002). [CrossRef]

, 2

2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

].

However, the current BLI technology has not explored the full potential of this approach. It only works in 2D imaging mode and is incapable of 3D imaging of the light source location associated with specific organs and tissues [2

2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

]. Since its first introduction in 2003 [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

], the bioluminescence tomography (BLT) has been undergone a rapid development [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

, 13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

, 14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

, 15

15. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

, 16

16. 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,” Academic Radiology 11, 1029–1038 (2004). [CrossRef] [PubMed]

, 17

17. X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12, 3996–4000 (2004). [CrossRef] [PubMed]

, 18

18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

, 19

19. 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 method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005). [CrossRef] [PubMed]

, 20

20. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857 (2005). [CrossRef] [PubMed]

, 21

21. C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

, 4

4. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005). [CrossRef] [PubMed]

, 22

22. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005). [CrossRef] [PubMed]

, 2

2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

, 23

23. N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006). [CrossRef] [PubMed]

, 24

24. H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005). [CrossRef]

]. BLT is to address the needs for 3D localization and quantification of an internal bioluminescent source distribution in a small animal. With BLT, optimal analyzes on a bioluminescent source distribution become feasible inside a living mouse. Although there are currently several approaches for this technique [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

, 13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

, 14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

, 15

15. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

, 16

16. 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,” Academic Radiology 11, 1029–1038 (2004). [CrossRef] [PubMed]

, 17

17. X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12, 3996–4000 (2004). [CrossRef] [PubMed]

, 18

18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

, 19

19. 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 method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005). [CrossRef] [PubMed]

, 20

20. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857 (2005). [CrossRef] [PubMed]

, 21

21. C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

, 4

4. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005). [CrossRef] [PubMed]

, 22

22. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005). [CrossRef] [PubMed]

, 2

2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

, 23

23. N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006). [CrossRef] [PubMed]

, 24

24. H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005). [CrossRef]

], the originally proposed approach in [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

] is still one of the promising techniques in this field. With this technique, the complete knowledge on the optical properties of anatomical components is assumed to be available from an independent tomography scan, such as a CT/micro-CT, MRI scan and/or diffuse optical tomography (DOT), by image segmentation and optical property mapping. That is, we can segment the image volume into a number of anatomical structures, and assign optical properties to each component using a database of the optical properties, or use the DOT technique for this purpose [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

].

case. Then we extend two of our reconstruction methods for BLT in [14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

, 15

15. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

, 18

18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

] to the partial measurement case. The first algorithm is a variant of the well-known expectation-maximization (EM) algorithm [32

32. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

, 33

33. L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982). [CrossRef] [PubMed]

, 34

34. D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992). [CrossRef]

]. The second one is based on the projected Landweber scheme [35

35. M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003). [CrossRef] [PubMed]

, 36

36. M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

, 37

37. M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997). [CrossRef]

, 38

38. A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998). [CrossRef]

]. Both methods allow the incorporation of knowledge-based constraints. Two practical constraints, the source non-negativity and support constraints, are introduced to regularize the BLT problem and produce stability. The initial choice of both methods and its influence on the regularization and stability are also discussed. Both algorithms are evaluated and validated with intensive numerical simulation and a physical phantom. Finally algorithmic issues are investigated, especially the method to avoid the inverse crime in numerical simulations.

The organization of the paper is as follows. We introduce the problem of BLT from partial measurement in §2, reformulate it in an operator form by generalizing the Dirichlet-to- Neumann map in §3, and extend our results on the solution uniqueness of BLT in §4. Then, we present our iterative BLT algorithms in §6. We report our numerical and experimental results in §7. Finally we discuss technical problems and research topics with the current BLT techniques in §8 and conclude the paper in §9.

2. Formulation of BLT

Let Ω be a bounded smooth domain in the three–dimensional Euclidean space R 3 that contains an object to be imaged. BLT is to reconstruct the source q from measurement on the boundary of Ω enclosing the source. Let u(x,θ) be the radiance in direction ϕS 2 at x∈Ω, where S 2 is the unit sphere. A general model for light migration in a random medium is the radiative transfer equation (RTE) [39

39. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

, 25

25. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999). [CrossRef]

, 26

26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

]

1cut(x,θ,t)+θ·xu(x,θ,t)+μ(x)u(x,θ,t)=μs(x)S2η(θ·θ)u(x,θ,t)dθ+q(x,θ,t)
(1)

for t>0, and x∈Ω, where c denotes the particle speed, µ=µa+µs with µa and µs being the absorption and scattering coefficients respectively, the scattering kernel η is normalized such that ∫S 2η(θ·θ′)′=1, and q is the internal light source. In (1), the radiance u(x,θ,t) is in Wcm-2 sr-1, the source term q(x,θ,t) in Wcm-3 sr-1, the scattering coefficient µs and the absorption coefficient µa both are given in cm-1, and the scattering phase function η is in sr-1 [40

40. A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003). [CrossRef]

]. To find a solution for (1), we need the initial condition

u(x,θ,0)=0,xΩ,θS2,
(2)

and the boundary condition for u

u(x,θ,t)=g(x,θ,t),xΓ,θS2,ν(x)·θ<0,t>0,
(3)

where ν is the exterior normal on the boundary Γ of Ω. The forward problem (1), (2) and (3) admits a unique solution under natural assumptions on µ, µa and η [41

41. D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

]. The homogeneous condition g-(x,θ,t)=0 specifies that no photons travel in an inward direction at the boundary, except for source terms [25, p. R50], which is the case for BLT. In terms of the radiance u(x,θ,t), the measurement at a boundary points is given by

g(x,t)=S2ν(x)·θu(x,θ,t)dθ,x=Γ,t0.
(4)

With the RTE, it is quite complex to reconstruct the light source q from the measurement g and with the above initial-boundary problem being as the forward process. This is largely due to the difficulty in computing the solution u for the forward problem (1), (2) and (3).

Typical values of the optical parameters for biological tissues are µa=0.1–1.0mm-1, µs= 100–200mm-1, respectively. This means that the mean free path of the particles is between 0.005 and 0.01 mm, which is very small compared to a typical object. Thus the predominant phenomenon in optical tomography is scattering rather than transport. Therefore, the diffusion approximation has been widely applied to simplify the RTE (1) in optical tomography [25

25. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999). [CrossRef]

, 26

26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

, 27

27. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef] [PubMed]

]. The diffusion approximation assumes that the radiance u(x,θ,t) is isotropic and takes the average radiance u0(x,t) to approximate the radiance u(x,θ,t)

u0(x,t)=14πS2u(x,θ,t)dθ.
(5)

Letη¯=14πS2θ·θη(θ·θ)dθand

μs=(1η¯)μs,
(6)
D(x)=13(μa(x)+μs(x)),
(7)
q0(x,t)=14πS2q(x,θ,t)dθ.
(8)

The forward process in the diffusion regime is then given by the following initial-boundary problem [39

39. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

, 25

25. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999). [CrossRef]

, 26

26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

]

1cu0(x,t)t·(D(x)u0(x,t))+μa(x)u0(x,t)=q0(x,t),xΩ,t>0
(9)
u0(x,0)=0,xΩ,
(10)
u0(x,t)+2D(x)u0ν(x,t)=g(x,t),xΓ,t>0.
(11)

For diffusion approximation based BLT, the measured quantity at x∈Γ for t≥0 is given by

g(x,t)=D(x)u0ν(x,t).
(12)

Because the internal bioluminescence distribution is relatively stable, we can use the stationary version of (9) and (11) as the forward model for BLT. By discarding all the time dependent terms in (9) and (11), the stationary forward model is given by the following boundary value problem (BVP)

·(D(x)u0(x))+μa(x)u0(x)=q0(x),xΩ,
(13)
u0(x)+2D(x)u0ν(x)=g(x),xΓ.
(14)

In practice, it is difficult to obtain all the measurement on the boundary Γ. We consider the case in which the measurement is conducted on some disjoint patches Γj⊂Γ for j=1, ⋯,J, each of which is smooth, closed and connected. Let ΓP=∪Jj=1 Γj. The BLT problem from partial measurement can be stated as follows: Given the incoming light g- on Γ and outgoing radiance g on ΓP, find a source q0 with the corresponding diffusion approximation u0 such that

BLT(P){·(Du0)+μau0=q0,inΩ,u0+2Du0ν=g,onΓ,Du0ν=g,onΓP.
(15)

In a typical BLT setting, g-=0, because there is no incoming light. The optical parameters D and µa in the above formulations can be established point-wise as mentioned above [11

11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

, 12

12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

].

3. Reformulation of BLT(P)

The uniqueness property of the BLT problem was already studied [13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

]. For the BLT(P) problem (15), we analyze its solution uniqueness in this section. The reader is reminded that the presentation in its mathematically accurate form will require rather technical and tedious assumptions on the domain and the coefficients. Here we will make the mathematical presentations as precise as possible while keeping a reasonable readability. For details, please refer to [13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

, 14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

] and the references therein.

In additions to the assumptions for the the BLT(P) problem (15), we assume that the parameters D>D0>0 for some positive constant D0 and that µa=0 are bounded functions. We further assume that D is sufficiently regular near Γ. For example, D is equal to a constant near Γ. Let γ0 and γ1 be the boundary value maps

γ0[u]=uΓ,andγ1[u]=DuνΓ
(16)

and L be the differential operator

L[u]=·(Du)+μau.
(17)

Given fH12(ΓP), let w 1H 1(Ω) be the solution of the following mixed boundary value problem (MBVP) [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

, 43

43. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

]

L[w1]=0,inΩ,
(18)
γ0[w1]=f,onΓP
(19)
γ0[w1]+2γ1[w1]=g,onΓΓP.
(20)

We define a linear operator NΓP from H12(ΓP) to H12(ΓP)P) by

NΓP[f]=γ1[w1]ΓP.
(21)

On the other hand, for q0L 2(Ω), we consider the following MBVP

L[w2]=q0,inΩ,
(22)
γ0[w2]=0,onΓP,
(23)
γ0[w2]+2γ1[w2]=0,onΓΓP.
(24)

and define another linear operator ΛΓP from L 2(Ω) to H12(ΓP) by

ΛΓP[q0]=γ1[w2]ΓP.
(25)

Both operators P and ΛΓP are extensions of the well-known Dirichlet-to-Neumann (or Steklov-Poincareé) map ([44]) and an relevant operator Λ in the complete measurement case ([14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

, 15

15. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

, 18

18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

]) to the partial measurement case, respectively.

Assume that q0 is a solution of the BLT(P) problem (15) with one corresponding radiance diffusion approximation solution u, given the observed g on G P and assumed g- on Γ. Then u is the unique solution of the following MBVP

L[u]=q0,inΩ,
(26)
γ0[u]=g+2g,onΓP,
(27)
γ0[u]+2γ1[u]=g,onΓΓP.
(28)

On the other hand, let w1 be defined as in (18)–(20) with f=g-+2g, w 2 be defined as in (22)–(24), and ν=w 1+w 2. Then we have

L[v]=q0,inΩ.
(29)
γ0[v]=g+2g,onΓP,
(30)
γ0[v]+2γ1[v]=g,onΓΓP.
(31)

By the solution uniqueness of this MBVP [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

], it follows that v satisfies (26)–(28). Hence, u=ν is the required radiance u that generates the measurement on ΓP. The measurement equation (12) implies that

g=γ1[u]=γ1[w1]+γ1[w2]=NΓP[g+2g]ΛΓP[q0],onΓP.
(32)

Hence, q0 satisfies the following equation

ΛΓP[q0]=NΓP[g+2g]+g,onΓP.
(33)

Conversely, if there exists a q0 satisfying (33), we can construct u=v=w1 +w2 as above. It follows that u satisfies the forward model and the measurement equation. In summary we have

Proposition 3.1. q0 is a solution to the (BLT(P)) problem (15) if and only if it is a solution to the equation (33).

4. Solution structure of BLT(P)

We need the following notations from functional analysis [45

45. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

]. Let A be a linear operator from a Banach space X to a Banach space Y. The kernel or null space of A is defined as 𝒩 [A]={xX : A[x]=0}, and the range of A is 𝓡[A]= {yY: y=A[x] for some xX}. For a subspace M of a Hilbert space H, M⊥ is the set of all yH, such that 〈y,x〉=0 for all xM.

By Theorem 3.1, the uniqueness of BLT(P) solution can be characterized by determining the kernel 𝒩[ΛΓP] of the operator ΛΓP:L2(Ω)=H12(ΓP)L2(ΓP). We need the Green’s formula [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

, 43

43. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

]

Ω[v·L[w]w·L[v]]dx=Γ[vγ1[w]wγ1[v]]dΓ.
(34)

For ψH12(ΓP), let

TΓP be defined by

ϕ=TΓP[ψ],
(35)

as the unique solution in H 1(Ω)⊂L 2(Ω) of the boundary problem

L[ϕ]=0,inΩ,
(36)
γ0[ϕ]=ψ,onΓP,
(37)
γ0[ϕ]+2γ1[ϕ]=0,onΓΓP.
(38)

Then, we can prove with the Green’s formula that for the operators ΛΓP:L2(Ω)H12(ΓP)L2(ΓP) and TΓP:H12(ΓP)L2(ΓP)L2(Ω),

q0,TΓP[ψ]L2(Ω)=ΛΓP[q0],ψL2(ΓP),
(39)

i.e., they are adjoint to each other,

ΛΓP*=TΓP.
(40)

Hence, the kernel of ΛΓP is [45

45. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

]

𝒩[ΛΓP]=𝓡[ΛΓP*]=𝓡[TΓP].
(41)

To obtain an explicit characterization of 𝒩[ΛΓP], let

H0,ΓP2(Ω)={pH2(Ω):γ0[p]ΓP=0,γ1[p]ΓP=0,andγ0[p]+2γ1[p]ΓΓP=0}.
(42)

Then we have

Proposition 4.1.

𝓡[TΓP]=L[H0,ΓP2(Ω)].
(43)

Proof. If qL[H0,ΓP2(Ω)] with q=L[p] for some pH0,ΓP2(Ω), then for v=TΓP[ψ]𝓡[TΓP], by the Green’s formula (34),

q,vL2(Ω)=ΩvL[p]dx=Γ[vγ1[p]pγ1[v]]dΓ+ΩL[v]pdx=0,

because γ0[p]ΓP=0, γ1[p]ΓP=0, and the boundary integral on Γ\ΓP is equal to zero. Hence, q𝓡[TΓP]. Therefore, L[H0,ΓP2(Ω)]𝓡[TΓP].

Conversely, assume that q𝓡[TΓP]=𝒩[ΛΓP]. We have, by (22)–(25), there exists w 2 such that

L[w2]=q,inΩγ0[w2]=0,onΓPγ0[w2]+2γ1[w2]=0,onΓ\ΓP,γ1[w2]=0,onΓP.

We have w2H 2(Ω) by the regularity theory for second order elliptic partial differential equations [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

, 43

43. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

]. The above boundary conditions imply that w2H0,ΓP2(Ω). Hence, q=L[w2]L[H0,ΓP2(Ω)]. The conclusion follows immediately.

By Proposition 3.1, the BLT(P) problem is equivalent to the linear equation (33) with q 0 as the unknown to be found. All the solutions q 0 to (33) form a convex set in L 2(Ω). Hence, there exists one unique solution of the minimal L 2-norm among those solutions [45

45. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

]. Let this minimal norm solution be denoted as qH. Then, all the solutions can be expressed as qH+𝒩[Λ].

We summarize the above results into the following theorem.

Theorem 4.2. Assume that the BLT(P) problem is solvable. For any couple (g-,g) such that

NΓP[g+2g]+gH12(ΓP),
(44)

there is one special solution qH for the BLT(P) problem (15), which is of the minimal L2-norm among all the solutions. Then, any solution can be expressed as q0=qH+L[p], for some p=H0,ΓP2(Ω).

5. Uniqueness of the BLT(P) solution in RBF

It is remarkable that the uniqueness results in [13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

] for sources consisting of radial base functions (RBF) still hold for the BLT(P) problem after examining the assumptions and proofs therein. We assume the following conditions on Ω, D, µa and q0.

C1: Ω is a bounded C 2 domain of R 3 and partitioned into non-overlapping sub-domains Ωi, i=1,2,…, I;

C2: Each Ωi is connected with piecewise C 2 boundary Γi;

C3: D and µa are C 2 near the boundary of each sub-domain.

C4: D>D 0>0 for some positive constant D 0 is Lipschitz on each sub-domain; µa≥0 and µaLp(Ω) for some p>N/2;

The first uniqueness result is for sources consisting of bioluminescent point impulses

q0(y)=s=1Sasδ(yys).
(45)

where each as is a constant, and ys the location of a point source inside some Ωi, for s=1,⋯,S.

Theorem 5.1. ([13]) Assume the conditions C1–C4 hold. If q0(y)=s=1Sasδ(yys) and q0(y)=s=1Sasδ(yys) are two solutions to the BLT(P) problem (15) with the same measurement on ΓP, then S=S′ and there is a permutation τ of {1,⋯,S} such that a′s=aτ(s) and y′s=yτ(s).

q0(y)=s=1Sgs(yxs)χBr0s,r1s(xs)
(46)

where each gsL2(Br0s,r1s(xs)) is continuous, the source centers {xs} are distinct, and each source support Br0s,r1s(xs)Ωi for some 1≤i≤I. For each 0≤r0<r1<∞, x 0∈R3, Br0,r1(x0) denotes a hollow ball specified by r 0<‖x-x 0‖ < r1 for r 0>0 or a solid ball specified by ‖x-x0‖ < r1 for r0=0. Let φ(r) be defined as follows. For µa=0,

φ(r)=1,
(47)

and for µa>0,

φ(r)=sinh(μaDr)μaDr.
(48)

For the second result we need to replace the condition C4 with the following condition C4* and a new condition C5:

C4*: D and µa are piecewise constant in the sense that there exist constants D 1,⋯,D I>0 and µ 1,⋯,µ 1=0 such that D(x)Di and µa(x)=µi,∀x∈ Ωi.

C5: For each sub-domain Ωm, 1≤mI, there exists a sequence of indices 1≤i 1,i 2,…, i kI with the following connectivity property: the intersection ΓPΓi1 contains a smooth C 2 open patch and ΓijΓij+1 contains a smooth C 2 open patch, for j=1,…,k-1, and Ωik=Ωm;

Note that the condition C4* is a special case of the condition C4. The second uniqueness result can be stated as follows.

Theorem 5.2. ([13

13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

]) Assume the conditions C1–C3, C4* and C5 hold. If q1(y)=s=1Sgs(yys)χBr0s,r1s(ys) and q2(y)=s=1SGs(yYs)χBR0s,R1s(Xs) are two solutions to the BLT(P) problem (15) with the same measurement on ΓP, then S=S′ and there exist a permutation τ of
{1, ⋯ ,S} and a map C: {1,⋯,S} → [1,I] such that Ys = yτ(s) ∈ ΩC(s) and

r0sr1srN1φC(s)(r)gs(r)dr=R0τ(s)R1τ(s)rN1φC(s)(r)Gτ(s)(r)dr,fors=1,,S,
(49)

where φj is given by (48) or (47) with D=Dj and µaj.

6. Reconstruction methods

Let b=NΓP[g+2g]+gH12(ΓP). By (33), the BLT solution q0 satisfies the following equation

ΛΓP[q0]=b.
(50)

The extended EM and CL iterative algorithms for solving the BLT(P) problem (50) are established in the following.

6.1. Method I: EM method

Based on the formulation in [26

26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

, 14

14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

], let

F[q0]=ΓP{blogΛΓP[q0]ΛΓP[q0]}dΓ,
(51)

which is the log likelihood function when the measured data b is subject to Poissonian distribution. By the maximum principle of elliptic partial differential equations [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

], it follows that ΑΓP[q0]0 and b≥0 if q0≥0, q0≠0, g≥0, g≠0 and g-≥0. We try to find a solution for the BLT(P) problem by performing the following optimization

argmaxFq00[q0].
(52)

We first assume that q0>0 is a minimizer of F. The case of q0≥0 can be handled similarly as the limiting case. We need to find the Frechet derivative of F. Let

f(t)=F[q0+tv],fortaround0,
(53)

where v is an arbitrary bounded function of L 2(Ω), and compute

ddtf(t)t=0=ΓP{b1ΛΓP[q0]1}ΛΓP[v]dΓ=ΩΛΓP*[bΛΓP[q0]1]vdx.

Hence, the Frechet derivative of F is

F[q0]=ΛΓP*[bΛΓP[q0]1]L2(Ω).
(54)

If q0>0 is a solution of (52), it follows that F′[q0]=0. The general case of q0=0 is given by the following Kuhn-Tucker condition [26

26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

]

q0·ΛΓP*[bΛΓP[q0]1]=0.
(55)

Let ϕ1=ΛΓP*[1]=TΓP[1], i.e., the solution of the following MBVP, by (40) and (36)–(38),

L[ϕ1]=0,inΩ,
(56)
γ0[ϕ1]=1,onΓP,
(57)
γ0[ϕ1]+2γ1[ϕ1]=0,onΓΓP.
(58)

It follows from the maximum principle of elliptic partial differential equations that 0 <ϕ 1=1 [46

46. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).

]. Because ΛΓP*=TΓP by (40), the Kuhn-Tucker condition (55) can be rewritten as

q0=1ϕ1q0·TΓP[bΛΓP[q0]].
(59)

Then we obtain the following EM formula for BLT from partial measurement

q0(n+1)=1ϕ1q0(n)·TΓP[bΛΓP[q0(n)]].
(60)

6.2. Method II: CL method

Assume that we have some prior knowledge about the source represented as a convex set

𝓒={q0:q0satisfiessomeconvexconstraints.},
(61)

which is a closed convex subset of L 2(Ω). Let P𝓒 be the orthogonal projection operator from L 2(Ω) to 𝓒. Then the CL scheme, or projected Landweber scheme, for solving (50), is given as follows

q0(n+1)=P𝓒{q0(n)+λnΛΓP*[bΛΓP[q0(n)]]},
(62)

where λn is a relaxation parameter and ΛΓP*=TΓP by (40). The convergence property of the CL scheme was studied in [47

47. B. Eicke, “Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren,” Thesis, Technischen Universität Berlin (1991).

, 48

48. B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992). [CrossRef]

] and improved in [37

37. M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997). [CrossRef]

, 38

38. A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998). [CrossRef]

]. Conditions for the relaxation parameter depend on the operator norm ΛΓP*ΛΓP=ΛΓP2 For example, 0<λn<2ΛΓP2 [37

37. M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997). [CrossRef]

, 38

38. A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998). [CrossRef]

]. The limit of q(n)0 is a solution to the constrained least-squares problem

argminq0𝓒12bΛΓP[q0]L2(Γ)2.
(63)

In this work, the non-negativity and source support constraints are applied. The non-negativity constraint implies that the source is non-negative and is utilized by setting the negative parts of iterated sources to be zero. The source support constraint assumes that the source is non-zero only within some sub-region Ω0 of the region Ω, and is utilized by setting the iterated sources to be zero outside Ω0. The choice of Ω0 is discussed in §6.3.2. Both constraints are applied after each iteration.

6.3. Relevant issues

6.3.1. Computational environment

A common feature of the proposed EM and CL methods is that they are of an iterative nature. At each step, they require the evaluation of the operators ΛΓP by (25) based on the MBVP (22)–(24) and TΓP by (35) based on the MBVP (36)–(38). Both MBVPs are of the same type can be solved with the finite element method (FEM) [49

49. S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.

]. In this work, both MBVPs are solved with the FEM software Comsol and Matlab. Note b in (50) for both methods can be computed by (21) after solving the MBVP (18)™(20). The computer is a Dell workstation, Precision 670, with dual Intel Xeon CPUs of main frequency 2.8GHz and 6GB memory. The operating system is Microsoft Windows XP Professional X64 Edition. Other details are reported in §7.

6.3.2. Choice of q(0)0

In all iterative image restoration methods, we have to initialize the process. We propose the following method to choose an initial guess q(0)0. By Green’s formula, let v=u and p=1 in (34), we have,

Ω[μauq0]dx=ΓgdΓ.
(64)

If we replace q0 with q(0)0, we obtain

Ωq0(0)dx=ΓgdΓ+Ωμaudx.
(65)

Because u=w1 by the the maximum principle of elliptic partial differential equations [42

42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

], we have

Ωq0(0)dxΓgdΓ+Ωμaw1dx.
(66)

In practice, the source q0 is compactly supported on a subset Ω0 inside Ω. We choose q0 such that it is equal to a positive constant in its support Ω0 and zero otherwise. Hence

q0(0)=Q0χΩ0
(67)

where Q0 is a constant, and χΩ0 (x)=1 on Ω0 and is zero otherwise. The support Ω0 of the source q0 is part of the prior knowledge, which was termed the permissible region in and could be inferred from the measured data [19

19. 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 method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005). [CrossRef] [PubMed]

]. Please refer to §8 for further discussions. Because only the data g on ΓP is available, the final estimate is

Q0Ω0ΓPgdΓ+Ωμaw1dx,
(68)

where |Ω0| is the volume of Ω0.

6.3.3. Convergence criteria

The convergence criteria for both algorithms may include (1) when the iteration number n reaches an assumed maximum number; (2) when the successive incremental |q0(n+1)-q(n) 0| is smaller than an assumed error level. In this work, the convergence criterion is by manually setting the iteration numbers to fixed numbers for both methods, respectively.

7. Experimental results

7.1. Numerical experiments

For inverse problems, numerical tests of reconstruction methods usually make use of simulated data from the numerical solution of the forward problem. One typical issue is coined as the inverse crimes in [50

50. D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.

]. This happens especially when insufficient rough discretization or the same discretization are used for the forward and inverse process, because “it is possible that the essential ill-posedness of the inverse problem may not be evident” [50, p. 304]. Hence, the results could be overly optimistic and unreliable. “Unfortunately, not all of the numerical reconstructions which have appeared in the literature meet with this obvious requirement” [50, p. 133]. As suggested, “it is crucial that the synthetic data be obtained by a forward solver which has no connection to the inverse solver under consideration” [50, p. 133].

One approach to avoid the inverse crime is to use different discretizations in the forward and the inverse process [50, p. 304]. Because our formulation of the reconstruction methods is analytical and independent of any discretization, we can use different finite elements (shape functions and meshes) in Comsol for the forward process and reconstruction algorithms, respectively. Moreover, we can change the mesh sizes with the adaptive mesh technique at each iteration step for the reconstruction algorithms to solve the MBVPs (22)–(24) and (36)–(38). During the iteration intermediate results at different meshes are interpolated to the required nodes with the built-in bilinear interpolation method in Comsol, when values at the nodes are required.

We have carries out intensive numerical simulation for various numerical phantoms. Various algorithmic settings such as the initial choices, iteration numbers, shape functions and meshes of the FEM have also been evaluated. Because the algorithms depend on multiple factors, the optimal setting is still under investigation. Nevertheless both methods can reliably reconstruct the sources in most settings. Due to the limited space, we report one representative result with the CL method in the following.

In this experiment, a numerical phantom with the same geometrical structure as the physical phantom in §7.2 is used. This is a cylindrical heterogeneous mouse chest phantom (Fig. 1(a)) of 30mm height and 30mm diameter. Its structure is shown in Fig. 1(b). Three sources of the form

qi(x)=AiχΩi(x),Ω={xxx0<r},
(69)

for i=1,2,3, are set up in this simulation, where Ωi is a ball centered at xi: Ωi={x : ‖x-xi‖<ri}. The radius ri are all set to 1mm. The sources are centered at x 1=(-0.9cm, 0.25cm, 0cm), x 2=(-0.9cm, -0.25cm, 0cm) and x 3=(0.9cm, 0.25cm, 0cm), with the intensity values A 1=25.1µW/cm3, A 2=23.3µW/cm3 and A 3=25.1µW/cm3, respectively. These intensity values are set according to the total source power of the physical phantom in §7.2 so that the total source power of each source is equal to one of the physical sources in §7.2. The optical coefficients of the phantom are set as in Table 1.

Table 1. Optical parameters for the phantom

table-icon
View This Table
| View All Tables

As discussed above, we use different finite elements (shape functions and meshes) for the forward process and inverse process, respectively. The information of the finite elements is

presented in Table 2, which is part of the output of the command meshinfo of Comsol.

Fig. 1. (a) A heterogeneous mouse phantom consisting of bone (B), heart (H), lungs (L), and muscle (M). (b) A cross-section through two luminescent sources in the left lung and another source in the right lung. The four arrows show the directions of the CCD camera for data measurement.

Table 2. Finite element information for the simulation

table-icon
View This Table
| View All Tables

Figure 2 shows the results obtained with the CL method. The initial support or the permissible region is set to Q0={(x,y,z) : 0.8<(x 2+y 2)1/2 <1.2, -0.15<z<0.15}. The relaxation coefficient λ is manually set to λ=20. The computational overhead for the cases of complete measurement and partial measurement is about the same. It takes about 6 hours for 70 iterations to get the results in Fig. 2. The figures are generated with the commands postplot, geomplot and meshplot of Comsol with manually adjusted parameters at different views, respectively.

In the case of the partial measurement from the front view, the source in the right lung close to the back view is not reconstructed, see Fig. 2 (c) and (d). This is reasonable because that source is far from the measurement surface in terms of the mean-free path. When complete measured data is used for reconstruction, this source could be reliably reconstructed and is shown in Fig. 2 (a) and (b). Quantitative results about the location accuracy are compiled into Table 3. The absolute error is defined by (xi,rxi)2+(yi,ryi)2+(zi,rzi)2. (xi,r,yi,r,zi,r) is the reconstructed center of each source and is estimated interactively from the reconstructed

source distribution from different views. The relative error is defined by the absolute error divided by xi2+yi2+zi2. The results demonstrate that the same location accuracy for the left two sources can be obtained with only the partial measurement in the front view.

Fig. 2. Reconstructed results by the CL method and a cross-section at z=0cm. (a) and (b) are results from data measured at the four views. (c) and (d) are from data measured at the front view only.

Table 3. Quantitative results for the reconstructed locations of the three sources at S1=(-0.90, 0.25, 0), S2=(-0.90, -0.25, 0) and S3=(0.90, 0.25, 0), respectively. The unit is cm.

table-icon
View This Table
| View All Tables

Another quantitative index for BLT is the reconstructed source power compared to its orig-inal value. As reported in the literature, the source power was estimated as the source integralq(x)dx of the source intensity over its support [19

19. 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 method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005). [CrossRef] [PubMed]

, 51

51. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007). [CrossRef]

]. Another approach for the source power estimation for a RBF source is based on the result of Theorem 5.2 by computing the source moment ∫φs(‖x-Xs‖)Gs(‖x-Xs‖)dx over its support, which is equal to 4πR0sR1sr2φs(r)Gs(r)dr. The reconstructed source intensities and sizes are estimated interactively from orthogonal views of the reconstructed source distributions. Table 4 presents the results of the reconstructed source integrals and its absolute and relative errors with the true value, respectively, for each source. Table 5 shows the corresponding results for the source moments. The differences of the results in both tables are discussed in §8.

Table 4. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 3. Their true values are 105.1, 97.4 and 105.1, respectively. The unit is nW.

table-icon
View This Table
| View All Tables

Table 5. Quantitative results for the reconstructed source moments of the sources. The sources are listed in the order as in Table 3. Their true values are 125.5, 116.5 and 125.5, respectively. The unit is nW.

table-icon
View This Table
| View All Tables

7.2. Physical experiment

the latter. By bending the plastic vial, the glass vial can be broken to mix the two solutions after which luminescent light was emitted. The particular dye in the chemical solution was for red light, and it could last for approximately 4 hours at an emission wavelength range between 650nm and 700nm, being close to that of the red spectral region of the luciferase. Two small holes of diameter 0.6mm and height 3mm were drilled in the phantom with their centers at (-0.9cm, 0.15cm, 0.0cm) and (-0.9cm, -0.15cm, 0.0cm) in the left lung region of the phantom, respectively. Two red luminescent liquid filled catheter tubes of 1.9mm height and 0.56mm diameter were placed inside the two small holes, respectively. We measured the total power of the red luminescent liquid filled polythene tubes with the CCD camera. They were 105.1 nW and 97.4 nW, respectively.

Fig. 3. (a) A cross-section through two hollow cylinders for hosting luminescent sources in the left lung. The four arrows show the direction of the CCD camera during data acquisition. (b) The measurements at the four views combined along the phantom side surface with unit µW/cm2

Due to the limited space, representative results from the physical phantom are given in Fig. 4(a) and Fig. 4(b) using the EM algorithm from only the data measured in the four views of the side surface of the phantom. We conducted experiments for reconstruction from the data measured in one or several of the four views. Figure 4(c) and 4(d) are the results reconstructed using the EM algorithm from only the data measured in the front view. The results from the measured data in other single views or combinations of these three views were not encouraging, because the signal-to-noise ratios were too low. For the results in Fig. 4 (a)–(d), the iteration number for the EM algorithm was set to 50 along with an initial source support region, Q0={(x,y,z) : 0.8<(x2+y2)1/2<1.2,-0.15<z<0.15}. It takes about 5 hours for the 50 iterations of the EM algorithm. For the CL algorithm, similar results were obtained but the separation of the two sources was not as good as that obtained using the EM algorithm.

Fig. 4. Representative results reconstructed by the EM algorithm. (a) and (c) are the sources reconstructed by the EMalgorithm from the data measured in the four views and in the front view only, respectively. (b) and (d) are cross-sections at z=0 cm of the sources in (a) and (c), respectively.

Quantitative results about the location accuracy and source power computed by the source integral are summarized in Table 6 and Table 7. The source powers estimated by source moments are not available because the original source is not of RBF sources, please refer to discussions in §8. The reconstructed source intensities and sizes are estimated interactively from orthogonal views of the reconstructed source distributions. The information of the finite element in this

experiment is presented in Table 8.

Table 6. Quantitative results for the reconstructed locations of the two sources at S1=(-0.90,0.15,0) and S2=(-0.90,-0.15,0), respectively. The unit is cm.

table-icon
View This Table
| View All Tables

Table 7. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 6. Their true values are 105.1 and 97.4, respectively. The unit is nW.

table-icon
View This Table
| View All Tables

Table 8. Finite element information for the physical phantom experiment

table-icon
View This Table
| View All Tables

8. Discussions

The iterative approach provides a mechanism for incorporating prior knowledge based constraints and has been widely used in practice. In this paper we have established two iterative algorithms for BLT from partial measurement. In this work, two constraints, the non-negativity and source support constraints, are applied. The EM algorithm induces the non-negativity of the source because of the maximum principle of elliptic partial differential equations, if the initial choice is non-negative. The source support with the EM algorithm is automatically implied because the support will not increase during the iteration due to its multiplication operation in (60). Hence, it seems that the EM algorithm is more preferable than the CL algorithm with those two constraints. However, the CL algorithm is more flexible to add other constraints and has the established convergence property. Other constraints highlighted by Theorem 5.1 and 5.2 are to restrict the solution space to a sub-space of bioluminescent source distributions so that the solution uniqueness holds to a practical extent. Nevertheless, the source support constraint helps to resolve the non-uniqueness because of the property of both the EM and CL algorithms.

There are remaining mathematical issues with the proposed algorithms. For the EM algorithm, its convergence has not been established, though it converges in our experiments. For the CL method, there is no guide in choosing the relaxation coefficient λn. This parameter depends on the operator norm ΛΓP*ΛΓP=ΛΓP2, which is equivalent to find the minimal eigenvalue of ΛΓP*ΛΓP or the minimal singular value of ΛΓP. This can be reduced to a boundary eigenvalue problem of partial differential equations. Both problems are left for future investigation.

There are also remaining physical issues with the proposed approach. We have used a geometric optics mapping of light beams to transform the measured data by a CCD detector to the surface measurement equation (12). More advanced techniques should be used to improve this process, such as the recently proposed non-contact measurement technique for fluorescence tomography [56

56. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Transactions on Medical Imaging 23, 492–500 (2004). [CrossRef] [PubMed]

]. Furthermore, the diffusion approximation can be improved by the radiative transfer equation [51

51. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007). [CrossRef]

].

9. Conclusions

Acknowledgments

Ming Jiang, Tie Zhou and Jiantao Cheng are supported in part by NKBRSF (2003CB716101) and NSFC (60325101, 60532080, 60628102), Chinese Ministry of Education (306017), Engineering Research Institute of Peking University, and Microsoft Research of Asia. Wenxiang Cong and Ge Wang are supported by NIH/NIBIB (EB001685), a special grant for bioluminescent imaging from Department of Radiology, College of Medicine, University of Iowa. The authors are grateful for anonymous referees for their important constructive comments.

References and links

1.

C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002). [CrossRef] [PubMed]

2.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005). [CrossRef]

3.

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001). [CrossRef] [PubMed]

4.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005). [CrossRef] [PubMed]

5.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004). [CrossRef] [PubMed]

6.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002). [CrossRef]

7.

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003). [CrossRef]

8.

A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003). [PubMed]

9.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003). [CrossRef] [PubMed]

10.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002). [CrossRef]

11.

G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

12.

G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

13.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004). [CrossRef] [PubMed]

14.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

15.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

16.

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,” Academic Radiology 11, 1029–1038 (2004). [CrossRef] [PubMed]

17.

X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12, 3996–4000 (2004). [CrossRef] [PubMed]

18.

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

19.

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 method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005). [CrossRef] [PubMed]

20.

A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857 (2005). [CrossRef] [PubMed]

21.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

22.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005). [CrossRef] [PubMed]

23.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006). [CrossRef] [PubMed]

24.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005). [CrossRef]

25.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999). [CrossRef]

26.

F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]

27.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef] [PubMed]

28.

A. Cong and G. Wang, “Multispectral bioluminescence tomography: Methodology and simulation,” International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.

29.

C. Q. Li and H. B. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004). [CrossRef] [PubMed]

30.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

31.

F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001). [CrossRef]

32.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

33.

L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982). [CrossRef] [PubMed]

34.

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992). [CrossRef]

35.

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003). [CrossRef] [PubMed]

36.

M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

37.

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997). [CrossRef]

38.

A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998). [CrossRef]

39.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

40.

A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003). [CrossRef]

41.

D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

42.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

43.

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

44.

V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Series (Springer, New York-Berlin-Heideberg, 1998).

45.

W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

46.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).

47.

B. Eicke, “Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren,” Thesis, Technischen Universität Berlin (1991).

48.

B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992). [CrossRef]

49.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.

50.

D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.

51.

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007). [CrossRef]

52.

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000). [CrossRef]

53.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).

54.

M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998). [CrossRef]

55.

R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications 236, 25–33 (1996). [CrossRef]

56.

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Transactions on Medical Imaging 23, 492–500 (2004). [CrossRef] [PubMed]

57.

M. D. Buhmann, Radial basis functions: theory and implementations, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2003). [CrossRef]

OCIS Codes
(170.0110) Medical optics and biotechnology : Imaging systems
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.6960) Medical optics and biotechnology : Tomography

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: April 9, 2007
Revised Manuscript: July 17, 2007
Manuscript Accepted: August 16, 2007
Published: August 20, 2007

Virtual Issues
Vol. 2, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Ming Jiang, Tie Zhou, Jiantao Cheng, Wenxiang Cong, and Ge Wang, "Image reconstruction for bioluminescence tomography from partial measurement," Opt. Express 15, 11095-11116 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-18-11095


Sort:  Year  |  Journal  |  Reset  

References

  1. C. Contag and M. H. Bachmann, "Advances in bioluminescence imaging of gene expression," Annu. Rev. Biomed. Eng. 4, 235 - 260 (2002). [CrossRef] [PubMed]
  2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotech. 23, 313 - 320 (2005). [CrossRef]
  3. B. W. Rice, M. D. Cable, and M. B. Nelson, "In vivo imaging of light-emitting probes," J. Biomed. Opt. 6, 432 - 440 (2001). [CrossRef] [PubMed]
  4. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, "Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study," Phys. Med. Biol. 50, 4225 -4241 (2005). [CrossRef] [PubMed]
  5. Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, "Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden," Molecular Imaging 3, 117-124 (2004). [CrossRef] [PubMed]
  6. A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, "Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging," Neoplasia 2, 491 - 495 (2002). [CrossRef]
  7. A. McCaffrey, M. A. Kay, and C. H. Contag, "Advancing molecular therapies through in vivo bioluminescent imaging," Molecuar Imaging 2, 75 - 86 (2003). [CrossRef]
  8. A. Soling and N. G. Rainov, "Bioluminescence imaging in vivo - application to cancer research," Expert Opinion on Biological Therapy 3, 1163 - 1172 (2003). [PubMed]
  9. J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, "Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography," Circulation 108, 1302 - 1305 (2003). [CrossRef] [PubMed]
  10. C. H. Contag and B. D. Ross, "It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology," J. Magn. Reson. 16, 378 - 387 (2002). [CrossRef]
  11. G. Wang, E. A. Hoffman, and G. McLennan, "Bioluminescent CT method and apparatus," (2003). US provisional patent application.
  12. G. Wang et al, "Development of the first bioluminescent tomography system," Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).
  13. G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems for bioluminescent tomography," Med. Phys. 31, 2289 -2299 (2004). [CrossRef] [PubMed]
  14. M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in "Proceedings of SPIE: Developments in X-Ray Tomography IV,", vol. 5535 (2004), vol. 5535, pp. 335 - 351. Invited talk.
  15. M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in "Proceedings of the RSNA," (2004).
  16. 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," Academic Radiology 11, 1029 - 1038 (2004). [CrossRef] [PubMed]
  17. X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, "Three-dimensional bioluminescence tomography with model-based reconstruction," Opt. Express 12, 3996-4000 (2004). [CrossRef] [PubMed]
  18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, "Image reconstruction for bioluminescence tomography," in "Proceedings of the RSNA," (2005).
  19. 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 method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005). [CrossRef] [PubMed]
  20. A. Cong and G. Wang, "A finite-element-based reconstruction method for 3D fluorescence tomography," Opt. Express 13, 9847-9857 (2005). [CrossRef] [PubMed]
  21. C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, "Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system," in "SMI Fourth Conference," (Cologne, Germany, 2005).
  22. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, "Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging," Phys. Med. Biol. 50, 5421 - 5441 (2005). [CrossRef] [PubMed]
  23. N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, "Iterative reconstruction method for light emitting sources based on the diffusion equation," Med. Phys. 33, 61 - 68 (2006). [CrossRef] [PubMed]
  24. H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, "Spectrally resolved bioluminescence optical tomography," Optics Letters 31, 365 - 367 (2005). [CrossRef]
  25. S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41 - R93 (1999). [CrossRef]
  26. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001). [CrossRef]
  27. A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005). [CrossRef] [PubMed]
  28. A. Cong and G. Wang, "Multispectral bioluminescence tomography: Methodology and simulation," International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.
  29. C. Q. Li and H. B. Jiang, "Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography," Opt. Express 12, 6313-6318 (2004). [CrossRef] [PubMed]
  30. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).
  31. F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001). [CrossRef]
  32. A. P. Dempster, N.M. Laird, and D. B. Rubin, "Maximal likelihood form incomplete data via the EM algorithm," Journal of the Royal Statistical Society. Series B. 39, 1 - 38 (1977).
  33. L. A. Shepp and Y. Vardi, "Maximum likelihood restoration for emission tomography," IEEE Transactions on Medical Imaging 1, 113 - 122 (1982). [CrossRef] [PubMed]
  34. D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, "Deblurring subject to nonnegativity constraints," IEEE Transactions on Signal Processing 40, 1143 - 1150 (1992). [CrossRef]
  35. M. Jiang and G. Wang, "Convergence studies on iterative algorithms for image reconstruction," IEEE Transactions on Medical Imaging 22, 569 - 579 (2003). [CrossRef] [PubMed]
  36. M. Jiang and G. Wang, "Development of iterative algorithms for image reconstruction," J. X-Ray Sci. Technol. 10, 77 - 86 (2002). Invited Review.
  37. M. Piana and M. Bertero, "Projected Landweber method and preconditioning," Inverse Problems 13, 441 - 463 (1997). [CrossRef]
  38. A. Sabharwal and L. C. Potter, "Convexly constrained linear inverse problems: iterative leat-squares and regularization," IEEE Transactions on Signal Processing 46, 2345 - 2352 (1998). [CrossRef]
  39. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).
  40. A. D. Klose and A. H. Hielscher, "Quasi-Newton methods in optical tomographic image reconstruction," Inverse Problems 19, 387-409 (2003). [CrossRef]
  41. D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).
  42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).
  43. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).
  44. V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Series (Springer, New York-Berlin-Heideberg, 1998).
  45. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.
  46. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).
  47. B. Eicke, "Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren," Thesis, Technischen Universit¨at Berlin (1991).
  48. B. Eicke, "Iteration methods for convexly constrained ill-posed problems in Hilbert space," Numerical Functional Analysis and Optimization 13, 413 - 429 (1992). [CrossRef]
  49. S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.
  50. D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.
  51. A. D. Klose, "Transport-theory-based stochastic image reconstruction of bioluminescent sources," J. Opt. Soc. Am., A 24, 1601-1608 (2007). [CrossRef]
  52. E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, "Inverse source problem and mimnimum-energy sources," J. Opt. Soc. Am., A 17, 34 - 45 (2000). [CrossRef]
  53. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).
  54. M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998). [CrossRef]
  55. R. J. Santos, "Equivalence of regularization and truncated iteration for general ill-posed problems," Linear Algebra and Its applications 236, 25-33 (1996). [CrossRef]
  56. R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental fluorescence tomography of tissues with noncontact measurements," IEEE Transactions on Medical Imaging 23, 492-500 (2004). [CrossRef] [PubMed]
  57. M. D. Buhmann, Radial basis functions: theory and implementations, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2003). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited