## Image reconstruction for bioluminescence tomography from partial measurement

Optics Express, Vol. 15, Issue 18, pp. 11095-11116 (2007)

http://dx.doi.org/10.1364/OE.15.011095

Acrobat PDF (734 KB)

### 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

*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. 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]

*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]

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

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]

*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]

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]

**4**, 235–260 (2002). [CrossRef] [PubMed]

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]

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]

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]

*bioluminescence tomography*(BLT) has been undergone a rapid development [11, 12, 13

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

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]

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]

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. 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**, 313–320 (2005). [CrossRef]

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]

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

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]

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]

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. 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**, 313–320 (2005). [CrossRef]

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]

*per se*. To obtain satisfactory results for the BLT, prior knowledge must be utilized to regularize the problem. The tomographic feasibility and the solution uniqueness have been theoretically studied in [13

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

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]

**50**, 4225–4241 (2005). [CrossRef] [PubMed]

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]

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]

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]

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

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

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

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]

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]

## 2. Formulation of BLT

**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

*ϕ*∈

*S*

^{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, 25

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]

*t*>0, and

*x*∈Ω, where c denotes the particle speed,

*µ*=

*µ*+

_{a}*µ*with

_{s}*µ*and

_{a}*µ*being the absorption and scattering coefficients respectively, the scattering kernel η is normalized such that ∫

_{s}*S*

^{2}η(

*θ·θ′*)

*dθ*′=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

*µ*and the absorption coefficient

_{s}*µ*both are given in cm

_{a}^{-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]

*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, µ*and η [41]. The homogeneous condition

_{a}*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

*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).

*µ*=0.1–1.0mm

_{a}^{-1},

*µ*= 100–200mm

_{s}^{-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]

*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]

*u(x,θ,t)*is isotropic and takes the average radiance

*u*to approximate the radiance

_{0}(x,t)*u(x,θ,t)*

**15**, R41–R93 (1999). [CrossRef]

*Mathematical Methods in Image Reconstruction* (SIAM, Philadelphia, PA, 2001). [CrossRef]

*x*∈Γ for

*t*≥0 is given by

_{j}⊂Γ for

*j*=1, ⋯,

*J*, each of which is smooth, closed and connected. Let Γ

*=∪*

_{P}*=1 Γ*

^{J}_{j}*. The BLT problem from partial measurement can be stated as follows:*

_{j}*Given the incoming light g- on Γ and outgoing radiance g on Γ*

_{P}, find a source q_{0}with the corresponding diffusion approximation u_{0}such that## 3. Reformulation of BLT(P)

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

*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

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

*P*) problem (15), we assume that the parameters

*D*>

*D*>0 for some positive constant

_{0}*D*and that

_{0}*µ*=0 are bounded functions. We further assume that

_{a}*D*is sufficiently regular near Γ. For example,

*D*is equal to a constant near Γ. Let

*γ*and

_{0}*γ*be the boundary value maps

_{1}*L*be the differential operator

*w*

_{1}∈

*H*

^{1}(Ω) be the solution of the following mixed boundary value problem (MBVP) [42, 43]

*P*) by

*q*∈

_{0}*L*

^{2}(Ω), we consider the following MBVP

*L*

^{2}(Ω) to

*NΓ*and ΛΓ

_{P}*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, 15, 18]) to the partial measurement case, respectively.*

_{P}*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

_{1}be defined as in (18)–(20) with

*f*=

*g*-+2

*g*,

*w*

_{2}be defined as in (22)–(24), and

*ν*=

*w*

_{1}+

*w*

_{2}. Then we have

*v*satisfies (26)–(28). Hence,

*u*=

*ν*is the required radiance

*u*that generates the measurement on Γ

*. The measurement equation (12) implies that*

_{P}*q*satisfies the following equation

_{0}*q*satisfying (33), we can construct

_{0}*u=v=w*as above. It follows that

_{1}+w_{2}*u*satisfies the forward model and the measurement equation. In summary we have

*q*.

_{0}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)

*P*) problem is expected to have at least one solution. Therefore, we will not discuss the existence of the BLT(

*P*) problem but focus on the uniqueness of the BLT(

*P*) solution. The uniqueness of the BLT solution was discussed in our previous study [13]. In what follows we will demonstrate how to extend the results to the BLT(

*P*) problem.

*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*]={

*x*∈

*X*:

*A*[

*x*]=0}, and the range of

*A*is 𝓡[

*A*]= {

*y*∈

*Y*:

*y*=

*A*[

*x*] for some

*x*∈

*X*}. For a subspace

*M*of a Hilbert space

*H, M*⊥ is the set of all

*y*∈

*H*, such that 〈

*y,x*〉=0 for all

*x*∈

*M*.

*P*) solution can be characterized by determining the kernel

*H*

^{1}(Ω)⊂

*L*

^{2}(Ω) of the boundary problem

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

*Proof*. If

*q*=

*L*[

*p*] for some

*is equal to zero. Hence,*

_{P}*w*

_{2}such that

_{2}∈

*H*

^{2}(Ω) by the regularity theory for second order elliptic partial differential equations [42, 43]. The above boundary conditions imply that

*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]. Let this minimal norm solution be denoted as

*qH*. Then, all the solutions can be expressed as

*qH*+𝒩[Λ].

*there is one special solution qH for the BLT(P) problem (15), which is of the minimal L*

^{2}-norm among all the solutions. Then, any solution can be expressed as q_{0}=q_{H}+L[p], for some## 5. Uniqueness of the BLT(P) solution in RBF

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

*P*) problem after examining the assumptions and proofs therein. We assume the following conditions on Ω,

*D*,

*µ*and

_{a}*q*.

_{0}*C*

^{2}domain of

**R**

^{3}and partitioned into non-overlapping sub-domains Ω

_{i},

*i*=1,2,…,

*I*;

*is connected with piecewise*

_{i}*C*

^{2}boundary Γ

*;*

_{i}*D*and

*µ*are

_{a}*C*

^{2}near the boundary of each sub-domain.

*D*>

*D*

_{0}>0 for some positive constant

*D*

_{0}is Lipschitz on each sub-domain;

*µ*≥0 and

_{a}*µ*∈

_{a}*L*(Ω) for some

^{p}*p*>

*N*/2;

*a*is a constant, and ys the location of a point source inside some Ω

_{s}*, for*

_{i}*s*=1,⋯,

*S*.

*([13]) Assume the conditions C1–C4 hold. If*q 0 ( y ) = ∑ s = 1 S a s δ ( y − y s ) and q 0 ′ ( y ) = ∑ s = 1 S ′ a s ′ δ ( y − y s ′ ) 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)}*i*≤I. For each 0≤

*r*<

_{0}*r*<∞,

_{1}*x*

_{0}∈R

^{3},

*r*

_{0}<‖

*x*-

*x*

_{0}‖ <

*r*for

_{1}*r*

_{0}>0 or a solid ball specified by ‖

*x-x*‖ <

_{0}*r*for

_{1}*r*=0. Let φ(r) be defined as follows. For

_{0}*µ*=0,

_{a}*µ*>0,

_{a}*D*and

*µ*are piecewise constant in the sense that there exist constants

_{a}*D*

_{1},⋯,

*D*

_{I}>0 and

*µ*

_{1},⋯,

*µ*

_{1}=0 such that

*D(x)*≡

*D*and

_{i}*µ*=

_{a}(x)*µ*,∀x∈ Ω

_{i}*.*

_{i}*, 1≤*

_{m}*m*≤

*I*, there exists a sequence of indices 1≤

*i*

_{1},

*i*

_{2},…,

*i*

_{k}≤

*I*with the following connectivity property: the intersection

*C*

^{2}open patch and

*C*

^{2}open patch, for

*j*=1,…,

*k*-1, and

*([13]) Assume the conditions C1–C3, C4* and C5 hold. If *q 1 ( y ) = ∑ s = 1 S g s ( ∥ y − y s ∥ ) χ B r 0 s , r 1 s ( y s ) and q 2 ( y ) = ∑ s = 1 S ′ G s ( ∥ y − Y s ∥ ) χ B R 0 s , R 1 s ( X s ) 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, ⋯ ,

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

*S*}

*and a map C*: {1,⋯,

*S*} → [1,

*I*]

*such that Y*=

_{s}*y*

_{τ(s)}∈ Ω

_{C(s)}

*and*

*where φ*.

_{j}is given by (48) or (47) with D=D_{j}and µ_{a}=µ_{j}## 6. Reconstruction methods

*P*) problem by solving the linear equation (33).

*q*satisfies the following equation

_{0}*P*) problem (50) are established in the following.

### 6.1. Method I: EM method

*Mathematical Methods in Image Reconstruction* (SIAM, Philadelphia, PA, 2001). [CrossRef]

*b*is subject to Poissonian distribution. By the maximum principle of elliptic partial differential equations [42], it follows that

*b*≥0 if

*q*≥0,

_{0}*q*≠0,

_{0}*g*≥0,

*g*≠0 and

*g*-≥0. We try to find a solution for the BLT(

*P*) problem by performing the following optimization

*q*>0 is a minimizer of

_{0}*F*. The case of

*q*≥0 can be handled similarly as the limiting case. We need to find the Frechet derivative of

_{0}*F*. Let

*v*is an arbitrary bounded function of

*L*

^{2}(Ω), and compute

*F*is

*q*>0 is a solution of (52), it follows that

_{0}*F*′[

*q*]=0. The general case of

_{0}*q*=0 is given by the following Kuhn-Tucker condition [26

_{0}*Mathematical Methods in Image Reconstruction* (SIAM, Philadelphia, PA, 2001). [CrossRef]

*ϕ*

_{1}=1 [46]. Because

### 6.2. Method II: CL method

*L*

^{2}(Ω). Let

*P*be the orthogonal projection operator from

_{𝓒}*L*

^{2}(Ω) to 𝓒. Then the CL scheme, or projected Landweber scheme, for solving (50), is given as follows

*λ*is a relaxation parameter and

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

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]

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]

*q*0 is a solution to the constrained least-squares problem

^{(n)}_{0}. The choice of Ω

_{0}is discussed in §6.3.2. Both constraints are applied after each iteration.

### 6.3. Relevant issues

#### 6.3.1. Computational environment

_{™}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}

*q*. By Green’s formula, let

^{(0)}_{0}*v=u*and

*p*=1 in (34), we have,

*q*with

_{0}*q*, we obtain

^{(0)}_{0}*u*=w

_{1}by the the maximum principle of elliptic partial differential equations [42], we have

*q*is compactly supported on a subset Ω

_{0}_{0}inside Ω. We choose

*q*such that it is equal to a positive constant in its support Ω

_{0}_{0}and zero otherwise. Hence

*Q*is a constant, and

_{0}*χΩ*=1 on Ω0 and is zero otherwise. The support Ω

_{0}(x)_{0}of the source

*q*is part of the prior knowledge, which was termed the permissible region in and could be inferred from the measured data [19

_{0}**13**, 6756–6771 (2005). [CrossRef] [PubMed]

*g*on Γ

*P*is available, the final estimate is

_{0}| is the volume of Ω

_{0}.

#### 6.3.3. Convergence criteria

*n*reaches an assumed maximum number; (2) when the successive incremental |

*q*-

_{0}^{(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

*inverse crimes*in [50]. 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].

^{™}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.

*i*=1,2,3, are set up in this simulation, where Ω

*is a ball centered at*

_{i}*x*: Ω

_{i}*={*

_{i}*x*: ‖

*x-*i‖<

_{x}*r*}. The radius

_{i}*r*are all set to 1mm. The sources are centered at

_{i}*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}=25.1

*µ*W/cm

^{3},

*A*

_{2}=23.3

*µ*W/cm

^{3}and

*A*

_{3}=25.1

*µ*W/cm

^{3}, 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.

^{™}.

*Q*={(

_{0}*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.

*x*) is the reconstructed center of each source and is estimated interactively from the reconstructed

_{i,r},y_{i,r},z_{i,r}*source integral*∫

*q(x)dx*of the source intensity over its support [19

**13**, 6756–6771 (2005). [CrossRef] [PubMed]

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

*source moment ∫φ*over its support, which is equal to

_{s}(‖x-X_{s}‖)G_{s}(‖x-X_{s}‖)dx### 7.2. Physical experiment

**13**, 6756–6771 (2005). [CrossRef] [PubMed]

**13**, 6756–6771 (2005). [CrossRef] [PubMed]

**13**, 6756–6771 (2005). [CrossRef] [PubMed]

*Q*: 0.8<(

_{0}={(x,y,z)*x*)1/2<1.2,-0.15<

^{2}+y^{2}*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.

## 8. Discussions

*P*) problem in the general case by Theorem 4.2, one may consider to utilize the minimal norm solution

*qH*as the solution of the BLT(

*P*) problem. The minimal norm solution

*qH*is unique and also called the minimal energy solution, and advocated in other applications [38

**46**, 2345–2352 (1998). [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]

*P*) problem in general. Because sources of compact supports are commonly encountered in practice, it can be proved in the same way that such sources cannot be found as the minimal norm solution for the BLT(

*P*) problem [14]. It has been reported that adequate prior knowledge must be utilized to obtain a physically favorable BLT solution [11, 12, 13

**31**, 2289–2299 (2004). [CrossRef] [PubMed]

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]

**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]

**50**, 4225–4241 (2005). [CrossRef] [PubMed]

**50**, 5421–5441 (2005). [CrossRef] [PubMed]

**31**, 365–367 (2005). [CrossRef]

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]

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]

*λ*. This parameter depends on the operator norm

_{n}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]

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

## References and links

1. | C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. |

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. |

3. | B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. |

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. |

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 |

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 |

7. | A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging |

8. | A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy |

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 |

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. |

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. |

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 |

17. | X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express |

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 |

20. | A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express |

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. |

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. |

24. | H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters |

25. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems |

26. | F. Natterer and F. Wuübbeling, |

27. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

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 |

30. | A. Kak and M. Slaney, |

31. | F. Natterer, |

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. |

33. | L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging |

34. | D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing |

35. | M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging |

36. | M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. |

37. | M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems |

38. | A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing |

39. | A. Ishimaru, |

40. | A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems |

41. | D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, |

42. | D. Gilbarg and N. S. Trudinger, |

43. | R. Dautray and J. L. Lions, |

44. | V. Isakov, |

45. | W. Rudin, |

46. | M. H. Protter and H. F. Weinberger, |

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 |

49. | S. C. Brenner and L. R. Scott, |

50. | D. L. Colton and R. Kress, |

51. | A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A |

52. | E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A |

53. | A. N. Tikhonov and V. Y. Arsenin, |

54. | M. Bertero and P. Boccacci, |

55. | R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications |

56. | R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Transactions on Medical Imaging |

57. | M. D. Buhmann, |

**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

- C. Contag and M. H. Bachmann, "Advances in bioluminescence imaging of gene expression," Annu. Rev. Biomed. Eng. 4, 235 - 260 (2002). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. McCaffrey, M. A. Kay, and C. H. Contag, "Advancing molecular therapies through in vivo bioluminescent imaging," Molecuar Imaging 2, 75 - 86 (2003). [CrossRef]
- A. Soling and N. G. Rainov, "Bioluminescence imaging in vivo - application to cancer research," Expert Opinion on Biological Therapy 3, 1163 - 1172 (2003). [PubMed]
- 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]
- 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]
- G. Wang, E. A. Hoffman, and G. McLennan, "Bioluminescent CT method and apparatus," (2003). US provisional patent application.
- G. Wang et al, "Development of the first bioluminescent tomography system," Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).
- G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems for bioluminescent tomography," Med. Phys. 31, 2289 -2299 (2004). [CrossRef] [PubMed]
- 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.
- M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in "Proceedings of the RSNA," (2004).
- 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]
- 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]
- 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).
- 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]
- A. Cong and G. Wang, "A finite-element-based reconstruction method for 3D fluorescence tomography," Opt. Express 13, 9847-9857 (2005). [CrossRef] [PubMed]
- 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).
- 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]
- 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]
- 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]
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- 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.
- 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]
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- M. Jiang and G. Wang, "Convergence studies on iterative algorithms for image reconstruction," IEEE Transactions on Medical Imaging 22, 569 - 579 (2003). [CrossRef] [PubMed]
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- 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]
- 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]

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