## An optimal permissible source region strategy for multispectral bioluminescence tomography

Optics Express, Vol. 16, Issue 20, pp. 15640-15654 (2008)

http://dx.doi.org/10.1364/OE.16.015640

Acrobat PDF (579 KB)

### Abstract

Multispectral bioluminescence tomography (BLT) attracts increasing more attention in the area of small animal studies because multispectral data acquisition could help in the 3D location of bioluminescent sources. Generally, BLT problem is ill-posed and *a priori* information is indispensable to reconstruction bioluminescent source uniquely and quantitatively. In this paper, we propose a spectrally solved bioluminescence tomography algorithm with an optimal permissible source region strategy. Being the most different from earlier studies, an optimal permissible source region strategy which is automatically selected without human intervention is developed to reduce the ill-posedness of BLT and therefore improves the reconstruction quality. Furthermore, both numerical stability and computational efficiency benefit from the strategy. In the numerical experiments, a heterogeneous phantom is used to evaluate the proposed algorithm and the synthetic data is produced by Monte Carlo method for avoiding the *inverse crime*. The results demonstrate the feasibility and potential of our methodology for reconstructing the distribution of bioluminescent sources.

© 2008 Optical Society of America

## 1. Introduction

*in vivo*. Hence, it becomes an increasingly important instrument for biomedical researchers to diagnose diseases, evaluate therapies, and facilitate drug development with small animals such as mouse models [1

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

4. G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express **14**, 7801–7809 (2006). [CrossRef] [PubMed]

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

4. G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express **14**, 7801–7809 (2006). [CrossRef] [PubMed]

6. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006). [CrossRef] [PubMed]

7. H. Zhao, T. C. Doyle, O. Coquoz, F. Kalish, B. W. Rice, and C. H. Contag, “Emission spectra of bioluminescent reporters and interaction with mammalian tissue determine the sensitivity of detection in vivo,” J. Biomed. Opt. **10**, 041210-1–9 (2005). [CrossRef]

*nm*to 750

*nm*, which can be detected using a sensitive low light imaging systems, typically based on charge-coupled device (CCD) camera. Bioluminescent signal observed on the surface of the small animal forms the basis for tomographic reconstruction of internal bioluminescent source. However, three-dimensional (3D) bioluminescent reconstruction from boundary data is not unique and a highly ill-posed inverse problem in the general case [5

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

*a priori*information has a quite effect on source reconstruction [5

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

*priori*information is the permissible source region strategy. So far,

*a priori*permissible source region strategy has been developed for BLT reconstruction [8

8. 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]

4. G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express **14**, 7801–7809 (2006). [CrossRef] [PubMed]

6. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006). [CrossRef] [PubMed]

8. 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]

9. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007). [CrossRef] [PubMed]

*a posteriori*permissible source region method has also been developed [9

9. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007). [CrossRef] [PubMed]

*a priori*information to improve BLT reconstruction [9

9. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007). [CrossRef] [PubMed]

11. O. Coquoz, T. L. Troy, D. Jekic-McMullen, and B. W. Rice, “Determination of depth of in vivo bioluminescent signals using spectral imaging techniques,” Proc. SPIE **4967**, 37–45 (2003). [CrossRef]

12. 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]

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

14. A. X. Cong and G. Wang, “Multispectral Bioluminescence Tomography: Methodology and Simulation,” Int. J. Biomed. Imaging **2006**, Article ID 57614, 7 pages, 2006. doi:10.1155/IJBI/2006/57614. [CrossRef]

15. H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. **31**: 365–367 (2006). [CrossRef] [PubMed]

16. G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of Multiview and multispectral Data,” Int. J. Biomed. Imaging, **2006**. Article ID 58601, 8 pages, 2006. doi:10.1155/IJBI/2006/58601. [CrossRef]

16. G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of Multiview and multispectral Data,” Int. J. Biomed. Imaging, **2006**. Article ID 58601, 8 pages, 2006. doi:10.1155/IJBI/2006/58601. [CrossRef]

**14**, 7801–7809 (2006). [CrossRef] [PubMed]

6. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006). [CrossRef] [PubMed]

8. 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]

16. G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of Multiview and multispectral Data,” Int. J. Biomed. Imaging, **2006**. Article ID 58601, 8 pages, 2006. doi:10.1155/IJBI/2006/58601. [CrossRef]

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

19. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement,” Opt. Express **15**, 11095–11116 (2007). [CrossRef] [PubMed]

19. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement,” Opt. Express **15**, 11095–11116 (2007). [CrossRef] [PubMed]

## 2. Methods

### 2.1. Formulation of BLT

*nm*–800

*nm*wavelength range, photon scattering predominates over photon absorption. When the bioluminescence imaging experiment is performed in a dark environment, the propagation of photons can be described by the steady-state diffusion equation and Robin boundary condition [8

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

20. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

21. S. R. Arridge, “Optical tomography in medical imaging,”Inverse Probl. **15**, 41–93 (1999). [CrossRef]

*λ*on tissue optical property into account, the following model is given [9

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

12. 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]

*R*

^{3}that contains an object to be imaged;

*∂*Ω is the corresponding boundary; Φ(

*x*,

*λ*) denotes the photon flux density [

*Watts*/

*mm*

^{2}];

*S*(

*x*,

*λ*) is the bioluminescent source density [

*Watts*/

*mm*

^{3}];

*µ*

_{a}(

*x*,

*λ*) is the absorption coefficient [

*mm*

^{-1}];

*D*(

*x*,

*λ*)=1/(3(

*µ*

_{a}(

*x*,

*λ*)+(1-

*g*)

*µ*

_{s}(

*x*,

*λ*))) is the optical diffusion coefficient,

*µ*

_{s}(

*x*,

*λ*) the scattering coefficient [

*mm*

^{-1}], and g the anisotropy parameter;

*ν*(

*x*) the unit outer normal on

*∂*Ω. Given the mismatch between the refractive indices

*n*for Ω and

*n*

^{′}for the external medium,

*A*(

*x*;

*n*,

*n*

^{′}) can be approximately represented:

*n*

^{′}is close to 1.0 when the mouse is in air;

*R*(

*x*) can be approximated by

*R*(

*x*) ≈ -1.4399

*n*

^{-2}+0.7099

*n*

^{-1}+0.6681+0.0636

*n*[22

22. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. **22**, 1779–1792 (1995). [CrossRef] [PubMed]

_{j}⊂

*∂*Ω,

*j*=1,2,…,

*M*, each is smooth and connected [19

19. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement,” Opt. Express **15**, 11095–11116 (2007). [CrossRef] [PubMed]

^{M}

_{j=1}γ

_{j}, the measured quantity is the outgoing flux density

*Q*(

*x*,

*λ*) on partial boundary γ and it can be expressed:

### 2.2. The optimal permissible source region strategy

*m*bands

*τ*

_{1},…,

*τ*

_{m}using appropriate filters, with

*τ*

_{l}=[

*λ*

_{l-1},

*λ*

_{l}),

*l*=1,2,…,

*m*-1,

*τ*

_{m}=[

*λ*

_{m-1},

*λ*

_{m}]. Here

*λ*

_{0}<

*λ*

_{1}<…<

*λ*

_{m}is a partition of the spectrum range. Based on the finite element theory [23], the weak solution of the flux density Φ(

*x*,

*τ*

_{l}) ∈

*H*

^{1}(Ω) on each band

*τ*

_{l}is given through the Eqs. (1) and (2):

*H*

^{1}(Ω) is the Sobelev space and Ψ(

*x*,

*τ*

_{l}) is an arbitrary piece-wise test function. In the framework of adaptive finite element analysis, let {

*T*

_{1},…,

*T*

_{k},…} be a sequence of nested triangulation of the given domain Ω based on adaptive mesh refinement, where the sequence gradually changes from coarse to fine along with the increase in

*k*[6

**14**, 8211–8223 (2006). [CrossRef] [PubMed]

*V*

_{k}are introduced on the discretized level

*T*

_{k}, satisfying

*V*

_{1}⊂ …

*V*

_{k}⊂ … …

*H*

^{1}(Ω). Now, we only consider the

*k*th discretized level which includes

*V*

_{k}. Then Eq. (5) can be simplified as following matrix form on the single-band

*τ*

_{l}[6

**14**, 8211–8223 (2006). [CrossRef] [PubMed]

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

*K*

_{k}(τ

_{l}),

*C*

_{k}(

*τ*

_{l}),

*B*

_{k}(

*τ*

_{l}),

*F*

_{k}(

*τ*

_{l}) are given by

*M*

_{k}(

*τ*

_{l})=

*K*

_{k}(

*τ*

_{l})+

*C*

_{k}(

*τ*

_{l})+

*B*

_{k}(

*τ*

_{l}), in view to

*M*

_{k}(

*τ*

_{l}) is a sparse positive definite matrix, we have:

*A*

_{k}(

*τ*

_{l})=

*M*

^{-1}

_{k}(

*τ*

_{l})

*F*

_{k}(

*τ*

_{l}), by substituting

*A*

_{k}(

*τ*

_{l}) into the Eq. (8), we obtain the Eq. (9):

*a priori*knowledge to improve the BLT reconstruction [6

**14**, 8211–8223 (2006). [CrossRef] [PubMed]

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

*τ*

_{l}can be determined by performing a beforehand spectral analysis, that is

*S*(

*τ*

_{l})=

*ω*(

*τ*

_{l})

*S*, where

*ω*(

*τ*

_{l}) ≥ 0 and ∑

^{m}

_{l =1}

*ω*(

*τ*

_{l}) ≈1,

*S*denotes the total photon density. At the coarsest level (

*k*=1), the whole reconstruction object Ω is assumed to be the permissible source solution

*S*

_{1}, taking into account the above spectral distribution, based on Eq. (9), we have:

*A*

^{T}is the transpose of

*A*and

*A*

^{T}

*A*is an

*S*

_{1}can be updated iteratively by:

*n*is iteration number and

*β*

_{n}is the gradient of each iteration,

*α*

_{n}is the step size, in order to get

*α*

_{n}, we construct a nonlinear least-squares optimization problem:

^{meas}

_{1}(

*τ*

_{l}) to substitute Φ

_{1}(

*τ*

_{l}), because the effect of noise is low originally. With the increase of the number of iterations, the permissible source region will become smaller and smaller. When ‖β

*‖*

_{n}≤

*δ*or iteration number

*n*>

*N*

_{max}, the iteration is terminated and the rough region of the optimal solution

*S** is obtained, that is Ω*. We name it the optimal permissible source region and the sketch is shown in Fig. 1. Otherwise,

*S*

_{1}is updated by the Eq. (11). Here,

*N*

_{max}is maximum iteration number and the experiential typical value for δ is between 10

^{-2}and 10

^{-5}. Generally, for

*N*

_{max}and

*δ*, we choose 500 and 5×10

^{-3}for practical use, respectively.

*a priori*information. Therefore, the nonnegative penalty is adopted during the iteration process.

### 2.3. Source reconstruction

*k*(

*τ*

_{l})=

*M*

^{-1}

_{ei }(

*τ*

_{l})

*F*

_{k}(

*τ*

_{l})

*S*

_{k}(

*τ*

_{l}), taking the linear relation between the unknown source value

*S*

_{k}and boundary measured points

*G*

_{k}(

*τ*

_{l}) can be established by deleting the rows of [

*M*

_{k}(

*τ*

_{l})

^{-1}

*F*

_{k}(

*τ*

_{l})] corresponding to unmeasured points. Incorporating the optimal permissible source region, we have:

*W*

_{k}is a diagonal matrix for selecting the permissible region, that is:

*k*=1,

*s*

_{k}(

*i*) is the optimal solution

*S*

^{*}and

*s*

^{max}

_{k}is its maximum.

*k*≥ 2,

*s*

_{k(i)}and

*s*

^{max}

_{k}are the reconstructed results prolonged from

*k*-1th level and the corresponding maximum, that is

^{k}

_{k}-1 is the ratio operator from

*k*-1 to

*k*,

*γ*

^{k}is the ratio factor,

*S*

_{k-1}is the reconstruction result on (

*k*-1)th level. By retaining the columns of

*G*

_{k}

*W*

_{k}corresponding to the permissible region

*S*

^{p}

_{k}, the final form of the linear system between the measurable boundary flux Φ

_{meas}

^{k}and

*S*is obtained:

^{p}_{k}*A*

_{k}is a severely ill-conditioned matrix because of the ill-posedness of BLT, the surface measured data is corrupted by noise, it is not practical to directly solve for

*S*

^{p}

_{k}from linear system (19). Then the following

*k*th objective function is established:

*s*

^{k}

_{inf}and

*s*

^{k}

_{suf}are the kth level lower and upper bounds of the source density; Λ is the weight matrix,

*V*Λ=

*V*

^{T}Λ

*V*;

*ρ*

_{k}the regularization parameter.

*S*

^{init}

_{k}is initial value at the

*k*th level. For BLT, Θ

*k*(

*S*

^{p}

_{k}) is a large-scale optimization problem with box-bound constrains, there-fore, a spectral projected gradient-based large-scale optimization algorithm is modified to solve the least square problem [9

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

25. E. G. Birgin and J. M. Martinez, “A box-constrained optimization algorithm with negative curvature directions and spectral projected gradients,” Computing Sup. **15**, 49–60 (2001). [CrossRef]

26. E. G. Birgin and J. M. Martinez, “Large-scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients,” Computational Optimization and Applications **23**, 101–125, (2002). [CrossRef]

*posteriori*error estimation and local mesh refinement, similar disposal methods are adopted according to [6

**14**, 8211–8223 (2006). [CrossRef] [PubMed]

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

*N*

^{i}

_{k}on each level are selected as switch condition indexes for triggering local mesh refinement. The discrepancy between the measured and computational boundary nodal flux data and the number

*k*of mesh refinement are used as stop criterion. Finally, the flow chart of the algorithm is shown in Fig. 2.

## 3. Numerical simulation

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

*mm*height and 10

*mm*radius was designed to evaluate the proposed algorithm [6

**14**, 8211–8223 (2006). [CrossRef] [PubMed]

*nm*–750

*nm*] can be divided into five discrete bins with steps of 50

*nm*. In the experiments, the optical properties of each component are assumed as

*priori*information and optical property parameters of each bin are compiled in Table 1 [9

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

*inverse crime*, the synthetic data was produced by molecular optical simulation environment (MOSE) which was developed using Monte Carlo method [27

27. 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,” Acad. Radiol. **11**, 1029–1038 (2004). [CrossRef] [PubMed]

*mm*radius and source intensity of 0.238

*nano*-

*Watts*/

*mm*

^{3}was centered at (-3,5,15) inside the right lung as shown in Fig. 3(a). In MOSE, the heterogeneous phantomwas discritized by triangular elements with an average element diameter of about 1

*mm*, shown in Fig. 3(b). In the simulation, the bioluminescent source of each band was sampled by 1.0×10

^{6}photons and the default initial guess

*S*

^{0}

_{1}for searching the optimal permissible source region was set 0.01. When

*k*=1, the ratio factor

*γ*

^{k}=0;

*k*> 1,

*γ*

^{k}was initially set 10

^{-1}for selecting permissible source region and was changed by multiplying a factor of 10.0 with the increase of level

*k*. In the optimization process,

*s*

^{k}

_{inf},

*s*

^{k}

_{suf}and

*S*

^{init}

_{k}are equal to 0.0, 10000 and 1.0×10

^{-5}, respectively. The stopping threshold ε

_{Φ}and the maximum number of mesh refinement

*K*

_{max}were set to 1.0×10

^{-8}and 3, respectively. Noted that in the reconstruction procedure, the initial discretization of phantom was coarse volumetric mesh shown in Fig. 3(c). In terms of the present condition, a single-view image technique is feasible for acquiring partial measured data. So in our simulations, the data only from the front view shown in Fig. 3(d) is used in case of partial measurement.

### 3.1. Experiment Results

*S*

^{0}

_{1}was used. As for switch condition indexes, the terminated gradient norm

*ε*

_{g}and the maximum iteration number

*N*

^{max}

_{k}were set to 1.0×10

^{-7}and 5000, respectively.

*ρ*

_{k}was 1.16×10

^{-9}at each level. First, using partial measured data in [500

*nm*,550

*nm*], the optimal permissible source region was established and the reconstruction was performed, corresponding result was shown in Fig. 4(a). Despite the use of optimal permissible source region, the monochromatic measurement-based tomographic reconstruction result was still undesirable no matter how we changed the initial guess

*S*

^{0}

_{1},

*δ*,

*ρ*

_{k}or maximum iteration number

*N*

_{max}. However, using multispectral partial measured data, BLT reconstruction showed preferable result with the proposed algorithm after one level refinement and the result was shown in Fig. 4(c). By contrast, when complete measured data from four views was used for selecting the optimal permissible source region, the corresponding result was demonstrated in Fig. 4(e).

*ξ*) are defined:

*ξ*=|

*S*

_{recon}-

*S*

_{real}|/

*S*

_{real}, where (

*x*,

*y*,

*z*) is the reconstructed center of source and (

*x*

_{0},

*y*

_{0},

*z*

_{0}) is the actual center of source.

*S*

_{recon}and

*S*

_{real}are reconstructed source density and actual source density, respectively, the unit is

*nano*-

*Watts*/

*mm*

^{3}. Quantitative results about both the location and density of source were shown in Table 2.

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

*a priori*information.

### 3.1.2. BLT reconstruction with noisy data

### 3.1.3. Consideration of different initial guesses

*S*

^{0}

_{1}). It is regarded as a criterion to test the stability of the proposed algorithm. Due to the limited space, representative results using different initial guesses of iterative approach are given. When the initial guess was 0.01, the result had been reported, shown in Fig. 4(c). When the initial guesses were set 0 and 0.1, the reconstructed results all indicated the algorithm was stable and robust to initial guess. The results were shown in Fig. 6 and Table 4. When the initial guess was equal to zero, the maximum reconstruction density was 0.142

*nano*-

*Watts*/

*mm*

^{3}. Although the density had a great difference with real density, the recovered center position of bioluminescence source was satisfactory, that was (-1.54,5.35,14.57).

### 3.1.4. Optical property errors consideration

**14**, 7801–7809 (2006). [CrossRef] [PubMed]

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

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

28. Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental Study on Bioluminescence Tomography with Multimodality Fusion,” Int. J. Biomed. Imaging **2007**, 86741 (2007). [CrossRef]

*a priori*information to reduce the ill-posedness of BLT and deal with the nonuniqueness of BLT [5

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

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

12. 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]

**52**, 4497–4512 (2007). [CrossRef] [PubMed]

29. V. Ntziachristos, A. H. Hielscher, A. G. Yodh, and B. Chance, “Diffuse Optical Tomography of Highly Heterogeneous Media,” IEEE Transactions on Medical Imaging **20**, 470–478 (2001). [CrossRef] [PubMed]

*mm*, which accounts for the capability for tolerating optical property errors.

*mm*radius and 0.238

*nano*-

*Watts*/

*mm*

^{3}power density were placed in the right lung with edge-to-edge distance of 2

*mm*. Reconstruction results were shown in Talbe 5. In the case of +50% error for all tissues, the center position of two sources both had an offset of about 2

*mm*, but the sources could be distinguished. However, when the error for all tissues is -50%, the reconstructed center position of a source had a large offset and the two sources could not be resolved. The effect of two sources with different edge-to-edge distances is necessary to further explore.

## 4. Discussions and conclusion

*inverse crime*[8

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

## Acknowledgments

## References and links

1. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

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

3. | G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence ct scanner,” Radiology |

4. | G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express |

5. | G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. |

6. | Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express |

7. | H. Zhao, T. C. Doyle, O. Coquoz, F. Kalish, B. W. Rice, and C. H. Contag, “Emission spectra of bioluminescent reporters and interaction with mammalian tissue determine the sensitivity of detection in vivo,” J. Biomed. Opt. |

8. | 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 |

9. | Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. |

10. | W. Han, W. Cong, and G. Wang, “Mathematical Study and Numirical Simulation of Multispectral Bioluminescence Tomography,” Int. J. Biomed. Imaging 2006, 54390 (2006). |

11. | O. Coquoz, T. L. Troy, D. Jekic-McMullen, and B. W. Rice, “Determination of depth of in vivo bioluminescent signals using spectral imaging techniques,” Proc. SPIE |

12. | 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. |

13. | G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. |

14. | A. X. Cong and G. Wang, “Multispectral Bioluminescence Tomography: Methodology and Simulation,” Int. J. Biomed. Imaging |

15. | H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. |

16. | G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of Multiview and multispectral Data,” Int. J. Biomed. Imaging, |

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

18. | 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. |

19. | M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement,” Opt. Express |

20. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. |

21. | S. R. Arridge, “Optical tomography in medical imaging,”Inverse Probl. |

22. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. |

23. | S. S. Rao, |

24. | P. E. Gill, W. Murray, and M. Wright, |

25. | E. G. Birgin and J. M. Martinez, “A box-constrained optimization algorithm with negative curvature directions and spectral projected gradients,” Computing Sup. |

26. | E. G. Birgin and J. M. Martinez, “Large-scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients,” Computational Optimization and Applications |

27. | 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,” Acad. Radiol. |

28. | Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental Study on Bioluminescence Tomography with Multimodality Fusion,” Int. J. Biomed. Imaging |

29. | V. Ntziachristos, A. H. Hielscher, A. G. Yodh, and B. Chance, “Diffuse Optical Tomography of Highly Heterogeneous Media,” IEEE Transactions on Medical Imaging |

30. | S. Holder, |

31. | J. K. Willmann, N. V. Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discovery |

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 16, 2008

Revised Manuscript: May 26, 2008

Manuscript Accepted: September 3, 2008

Published: September 19, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Jinchao Feng, Kebin Jia, Guorui Yan, Shouping Zhu, Chenghu Qin, Yujie Lv, and Jie Tian, "An optimal permissible source region strategy for multispectral bioluminescence tomography," Opt. Express **16**, 15640-15654 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15640

Sort: Year | Journal | Reset

### References

- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, "Looking and listening to light: the evolution of wholebody photonic imaging," Nat. Biotechnol. 23, 313-320 (2005). [CrossRef] [PubMed]
- C. H. Contag and M. H. Bachmann, "Advances in bioluminescence imaging of gene expression," Annu. Rev. Biomed. Eng. 4, 235-260 (2002).
- G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, "Development of the first bioluminescence ct scanner," Radiology 229(P), 566 (2003).
- G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, "In vivo mouse studies with bioluminescence tomography," Opt. Express 14, 7801-7809 (2006). [CrossRef] [PubMed]
- G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289- 2299 (2004). [CrossRef] [PubMed]
- Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, "A multilevel adaptive finite element algorithm for bioluminescence tomography," Opt. Express 14, 8211-8223 (2006). [CrossRef] [PubMed]
- H. Zhao, T. C. Doyle, O. Coquoz, F. Kalish, B. W. Rice, and C. H. Contag, "Emission spectra of bioluminescent reporters and interaction with mammalian tissue determine the sensitivity of detection in vivo," J. Biomed. Opt. 10, 041210-1-9 (2005). [CrossRef]
- 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]
- Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, "Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation," Phys. Med. Biol. 52, 4497-4512 (2007). [CrossRef] [PubMed]
- W. Han, W. Cong, and G. Wang, "Mathematical Study and Numirical Simulation of Multispectral Bioluminescence Tomography," Int. J. Biomed. Imaging 2006, 54390 (2006).
- O. Coquoz, T. L. Troy, D. Jekic-McMullen, and B. W. Rice, "Determination of depth of in vivo bioluminescent signals using spectral imaging techniques," Proc. SPIE 4967, 37-45 (2003). [CrossRef]
- 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]
- G. Alexandrakis, F.R. Rannou, and A. F. Chatziioannou, "Tomographic bioluminescence imaging by use of a combined optical (OPET) system: a computer simulation feasibility study," Phys. Med. Biol. 50, 4225-4241 (2005). [CrossRef] [PubMed]
- A. X. Cong and G. Wang, "Multispectral Bioluminescence Tomography: Methodology and Simulation," Int. J. Biomed. Imaging 2006, Article ID 57614, 7 pages, 2006. doi:10.1155/IJBI/2006/57614. [CrossRef]
- H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen and M. S. Patterson, "Spectrally resolved bioluminescence optical tomography," Opt. Lett. 31: 365-367 (2006). [CrossRef] [PubMed]
- G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, "The first bioluminescence tomography system for simultaneous acquisition of Multiview and multispectral Data," Int. J. Biomed. Imaging, 2006. Article ID 58601, 8 pages, 2006. doi:10.1155/IJBI/2006/58601. [CrossRef]
- X. Gu, Q. Zhang, L. Larcom, and H. Jiang, "Three-dimensional bioluminescence tomography with model-based reconstruction," Opt. Express 12, 3996-4000 (2004). [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).
- M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, "Image reconstruction for bioluminescence tomography from partial measurement," Opt. Express 15,11095-11116 (2007). [CrossRef] [PubMed]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modeling photon transport in tissue," Med. Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
- S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, 41-93 (1999). [CrossRef]
- M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, "The finite element method for the propagation of light in scattering media: Boundary and source conditions," Med. Phys. 22, 1779-1792 (1995). [CrossRef] [PubMed]
- S. S. Rao, The Finite Element Method in Enginering, (Butterworth-Heinemann, Boston, 1999).
- P. E. Gill, W. Murray, and M. Wright, Practical optimization, (Academic Press, New York, 1981).
- E. G. Birgin and J. M. Martinez, "A box-constrained optimization algorithm with negative curvature directions and spectral projected gradients," Computing Sup. 15, 49-60 (2001). [CrossRef]
- E. G. Birgin, J. M. Martinez, "Large-scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients," Comput. Optim. Appl. 23, 101-125, (2002). [CrossRef]
- 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," Acad. Radiol. 11, 1029-1038 (2004). [CrossRef] [PubMed]
- Y. Lv, J. Tian, W. Cong, and G. Wang, "Experimental Study on Bioluminescence Tomography with Multimodality Fusion," Int. J. Biomed. Imaging 2007, 86741 (2007). [CrossRef]
- V. Ntziachristos, A. H. Hielscher, A. G. Yodh, and B. Chance, "Diffuse Optical Tomography of Highly Heterogeneous Media," IEEE Trans. Med. Imaging 20, 470-478 (2001). [CrossRef] [PubMed]
- S. Holder, Electrical Impedance Tomography (Institute of Physics Publishing, Bristol and Philadelphia, 2005).
- J. K. Willmann, N. V. Bruggen, L. M. Dinkelborg and S. S. Gambhir, "Molecular imaging in drug development," Nat. Rev. Drug Discovery 7, 591-607 (2008). [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.

« Previous Article | Next Article »

OSA is a member of CrossRef.