## A source reconstruction algorithm based on adaptive *hp*-FEM for bioluminescence tomography

Optics Express, Vol. 17, Issue 17, pp. 14481-14494 (2009)

http://dx.doi.org/10.1364/OE.17.014481

Acrobat PDF (1085 KB)

### Abstract

As a novel modality of molecular imaging, bioluminescence tomography (BLT) is used to *in vivo* observe and measure the biological process at cellular and molecular level in small animals. The core issue of BLT is to determine the distribution of internal bioluminescent sources from optical measurements on external surface. In this paper, a new algorithm is presented for BLT source reconstruction based on adaptive *hp*-finite element method. Using adaptive mesh refinement strategy and intelligent permissible source region, we can obtain more accurate information about the location and density of sources, with the robustness, stability and efficiency improved. Numerical simulations and physical experiment were both conducted to verify the performance of the proposed algorithm, where the optical data on phantom surface were obtained via Monte Carlo simulation and CCD camera detection, respectively. The results represent the merits and potential of our algorithm for BLT source reconstruction.

© 2009 OSA

## 1. Introduction

*in vivo*[1

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

4. T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. **17**(5), 545–580 (2003). [CrossRef] [PubMed]

6. R. Weissleder, “Scaling down imaging: molecular mapping of cancer in mice,” Nat. Rev. Cancer **2**(1), 11–18 (2002). [CrossRef] [PubMed]

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

7. J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. **27**(5), 48–57 (2008). [CrossRef] [PubMed]

*ill-posed*problem and the uniqueness research of BLT shows that

*a priori*information has a great influence on source reconstruction [8

8. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [CrossRef] [PubMed]

*a priori*information includes the optical parameters, the structure of small animals and the permissible source region. Optical parameters (absorption coefficient, scattering coefficient and anisotropy factor) of arbitrary tissue can be assigned from an optical database, or determined by diffuse optical tomography [9

9. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6756. [CrossRef] [PubMed]

10. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. **50**(12), 2837–2858 (2005). [CrossRef] [PubMed]

*A priori*permissible source region can be estimated by the surface photon flux distribution and the heterogeneous structure of the detected object [11

11. 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**(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]

12. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express **16**(20), 15640–15654 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-20-15640. [CrossRef] [PubMed]

9. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6756. [CrossRef] [PubMed]

*h*-finite element method (

*h*-FEM) was also used in BLT for its high performance [11

11. 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**(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]

12. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express **16**(20), 15640–15654 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-20-15640. [CrossRef] [PubMed]

*hp*-finite element method (

*hp*-FEM) in BLT. The

*hp*-FEM is a modern version of FEM which varies both the diameter and polynomial degree of elements in order to maximize the convergence rates. This method was first introduced in the 1980’s [13

13. M. Ainsworth and B. Senior, “Aspects of an adaptive *hp*-finite element method: Adaptive strategy conforming approximation and efficient solvers,” Comput. Methods Appl. M **150**(1-4), 65–87 (1997). [CrossRef]

*h*and polynomial degree of elements

*p*,

*hp*-FEM can arrive in an unconditional exponential convergence, which is superior to other numerical methods [14

14. M. Ainsworth, “A preconditioner based on domain decomposition for *hp*-finite element approximation on quasi-uniform meshes,” SIAM J. Numer. Anal. **33**(4), 1358–1376 (1996). [CrossRef]

*hp*-FEM algorithm employs the initial permissible source region as

*a priori*knowledge to establish a direct linear relationship between the unknown source variable and the known measured data. The

*hp*-FEM algorithm begins on an initial coarse volumetric mesh to recover the source distribution. Based on the solution on the coarse mesh, we choose appropriate

*p*- or

*h*-refinement strategy for each element in solution region. Several experiments were conducted to validate the proposed algorithm. First, we reconstruct the source with the synthetic data generated through a modified molecular optical simulation environment (MOSE) [15

15. 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**(9), 1029–1038 (2004). [CrossRef] [PubMed]

16. MOSE, http://www.mosetm.net/*.*

*hp*-FEM is presented in section 2. In section 3, we evaluate the performance of the proposed algorithm through numerical simulations and physical experiment. Discussions are given in the last section.

## 2. Method

### 2.1. Diffusion approximation and boundary condition

11. 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**(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]

17. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express **13**(24), 9847–9857 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-24-9847. [CrossRef] [PubMed]

*Ω*is the region of interest;

*x*[

*W*/

*mm*

^{2}];

*W*/

*mm*

^{3}];

*mm*

^{−1}];

*mm*

^{−1}] and

*g*is the anisotropy parameter [18–20

20. M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. **9**(6), 1336–1346 (2004). [CrossRef] [PubMed]

21. 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**(11), 1779–1792 (1995). [CrossRef] [PubMed]

*n*for

*Ω*and

21. 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**(11), 1779–1792 (1995). [CrossRef] [PubMed]

21. 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**(11), 1779–1792 (1995). [CrossRef] [PubMed]

### 2.2. Reconstruction method based on adaptive hp-FEM

*hp*-FEM, let

*k*th mesh level, the continuous field

*Ω*.where

*p*is the order of the interpolation basis functions,

*N*denotes the number of the interpolation basis functions,

*p*,

*i*th nodal value on the

*k*th mesh level. Similarly, the source

*k*th mesh level, respectively. The selection of interpolation basis functions

9. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6756. [CrossRef] [PubMed]

*a priori*knowledge. Thus, Eq. (9) can be reduced to: Equation (11) can be rewritten as:where

*ill-posed*nature of BLT [8

8. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [CrossRef] [PubMed]

*Λ*is the weight matrix, and

*λ*denotes the regularization parameter, and

*l*

_{2}norm penalty function. In this paper,A modified Newton method with active set strategy is employed to deal with the minimization problem [24].

*h*-refinement or

*p*-refinement for each tetrahedron element [13

13. M. Ainsworth and B. Senior, “Aspects of an adaptive *hp*-finite element method: Adaptive strategy conforming approximation and efficient solvers,” Comput. Methods Appl. M **150**(1-4), 65–87 (1997). [CrossRef]

*C*,

*h*and

*p*; the parameter

*t*depends on the regularity of the exact solution and is large when the solution is smooth [14

14. M. Ainsworth, “A preconditioner based on domain decomposition for *hp*-finite element approximation on quasi-uniform meshes,” SIAM J. Numer. Anal. **33**(4), 1358–1376 (1996). [CrossRef]

*h*-FEM has been already employed for BLT reconstruction [11

**14**(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]

*h*-refinement in the solution region. But compare with

*hp*-FEM, only linear interpolation basis functions are used in

*h*-FEM, which would induce poorer solution precision. The

*hp*-FEM can reduce the mesh size (

*h*-refinement) and increase the order interpolation basis functions (

*p*-refinement) on each mesh level synchronously, so a faster convergence rate can be obtained, which is also the rationale that we use adaptive

*hp*-FEM for BLT reconstruction.

*hp*-FEM strategy, according to the source distribution solved on the previous mesh level, some tetrahedron elements are chosen to be refined. When the refinement elements are inside the solution region, the

*p*-refinement is more effective than h-refinement due to higher order interpolation basis functions [25]. The major aim in this paper is to obtain the reconstruction results of source distribution, so we only consider the element in

*the priori*permissible source region, which is benefit to improve the result quality.

*hp*-Refinement criterion: let

*i*th tetrahedron of

*the priori*permissible source region, and

*β*is a constant,

*p*-refinement is performed on this tetrahedron, and

*h*-refinement is selected on the surrounding tetrahedron.

*p*= 1 or 2, and divide a selected tetrahedron into eight son tetrahedra for

*p*= 2 [25]. Following the

*p*-refinement, the

*h*-refinement divides a tetrahedron into 2 or 4 son tetrahedra. The

*p*-refinement and all the possibility of

*h*-refinement of a tetrahedron are shown in Fig. 1(a) and Fig. 1(b)-(d) respectively.

*hp*-FEM algorithm, we select the norm of the gradient

*k*th mesh level to the (

*k*+ 1)th mesh level, where

*hp*-FEM algorithm to decrease the

*ill-posedness*of inverse problem in source reconstruction. The initial permissible source region is given as a prior knowledge artificially. Then the permissible source region on the next mesh level is specified based on the reconstructed source on the current mesh level. If a tetrahedron satisfies the following condition, it will be sorted out to form the new permissible source region.where

*k*+ 1)th refinement mesh level,

*j*th tetrahedron element in

*k*th refinement mesh level. This strategy can select an optimal permissible source region on each mesh level, which can decrease the

*ill-posed*of BLT effectively, and obtain a better reconstruction result. The flow chart of the algorithm is shown in Fig. 2 .

## 3. Experiments and Results

### 3.1. Numerical simulations

*mm*height and 10

*mm*radius was applied to model a mouse chest. It consists of four ellipsoids and one cylinder to represent muscle, lungs, heart, bone and liver, as shown in Fig. 3 (a) . Optical parameters are listed in Table 1 . The parameters calculated by optical tomography procedure are corresponding to the physical materials, which are supported by Prof. Ge Wang's lab (Bioluminescence Tomography Laboratory, Department of Radiology, University of Iowa).

*inverse crime*, MOSE was used to obtain the synthetic data. It is difficult to obtain all the surface data of a cylinder phantom in actual measurement, so only the data on cylinder side is used for source reconstruction in this paper.

*mm*radius and 0.238

*nW/mm*power density was centered at (3, 5, 15) inside the right lung as shown in Fig. 3(a), and the whole right lung was specified as

^{3}*a priori*permissible source region. In the reconstruction procedure, a coarse volumetric mesh shown in Fig. 3(b) was chosen as the initial discretization of phantom. We set the lower bound

*h*-FEM and

*hp*-FEM. Their reconstruction results are shown in Fig. 4(a)-(c) , and Fig. 4(d)-(f) denote their corresponding cross sections. For FEM algorithm, the normal mesh used for reconstruction contains 4436 nodes; the ultima refined mesh is 3984 nodes for

*hp*-FEM, and 3945 nodes for

*h*-FEM. The differences of reconstruction results can be distinguished intuitionally from Fig. 4, and our proposed algorithm has a better reconstructed location and power density for the actual source than FEM and

*h*-FEM. In order to analyze the results quantitatively, we define the distance error

*mm*, and the relative source density error is less than 8%, which is far better than that using of the other two algorithms.

^{7}photons in this paper). Therefore, Gaussian noise with different levels is added to the synthetic data to evaluate the stability and robustness of the proposed algorithm. The noise is added by the flowing formula:where

*δ*is noise level parameter,

*E*is a random error generated by a MATLAB function

*randn*, and its mean value is

*n*is the number of surface measured nodes. This noise is as similar as the noise caused by the dark current of CCD camera in physical experiment.

*hp*-FEM. So their mesh size in the permissible source region is similar. Using FEM on fine grid, the BLT reconstruction program coded in MATLAB takes 942.26 seconds on our desktop computer (Intel(R) Core(TM) 2 CPU 6300 @ 1.86GHz and 2G RAM). The results are shown in Fig. 5 (a)-(b). However, the proposed algorithm only cost 254.01 seconds, and the results are shown in Fig. 5(c)-(d). It seems obvious that the reconstructed location using our proposed algorithm is better than that using FEM on a fine grid. And the quantitative comparison results are listed in Table 4 . The two reconstructed density using FEM on a fine grid is very similar, but the relative source density errors are both not ideal (16.39% and 21.37%). Furthermore, the reconstructed locations are very poor (the distance errors are 3.27

*mm*and 3.20

*mm*respectively). The reconstructed locations using

*hp*-FEM is much better (the distance errors are 0.14

*mm*and 0.66

*mm*, respectively), and the relative source density error of one source is 4.62%, which is also better than that using FEM on a fine grid. But the deficiency is that we cannot obtain a fine reconstructed density for the other source.

### 3.2. Physical experiment

*hp*-FEM are 1598 and 1237, respectively. The point coordinate of maximum photon flux density on the surface is (22.47, 1.25, 27.69). Because the phantom is homogeneous, we first set the permissible source region as

*mm*and 2.98

*mm*, respectively. Furthermore, a larger source permissible region

*mm*). Due to the adaptive

*h*-refinement and

*p*-refinement, the reconstruction result using the proposed algorithm is as similar as that obtained with the source permissible region

*P*

_{1.}The quantitative analysis of reconstructed location is listed in Table 5 . But the quantitative density reconstruction is not provided in this part. The reconstruction results show that the change of permissible source region has little influence on

*hp*-FEM but much influence on FEM.

## 4. Discussion and conclusions

*h*) and polynomial degree (

*p*) of elements are two important factors in adaptive FEM for BLT source reconstruction. The global reducing mesh size or increasing polynomial degree of elements is infeasible for the overwhelming computational complexity. Thus, the adaptive strategy is essential. In literature [11

**14**(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]

*h*-FEM algorithm to achieve a better reconstructed location and density of the source. In this paper, we developed a novel adaptive

*hp-*FEM based algorithm to reconstruct the bioluminescent source inside the phantom, and then evaluated its performance in numerical simulations and physical experiment, respectively. The main motivation for the use of

*hp*-FEM is inspired by the following result: ‘an optimal sequence of

*hp*-grids can achieve exponential convergence for elliptic problems with a piecewise analytic solution, whereas

*h*- or

*p*-FEM converge at best algebraically’ [27

27. I. Babuška and B. Guo, “Approximation properties of the *hp*-version of the finite element method,” Comput. Methods Appl. Mech. Eng. **133**(3-4), 319–346 (1996). [CrossRef]

*hp*-FEM algorithm and an intelligent permissible source region strategy, a more accurate reconstructed location and density of source can be obtained satisfyingly.

*l*

_{2}norm [8

8. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [CrossRef] [PubMed]

*l*

_{2}norm, one of the reconstructed sources is weaker than the actual source in dual source reconstruction. Another Tikhonov regularization method based on

*l*

_{1}norm has been reported in literature [29

29. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**(10), 8062–8080 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-10-8062. [CrossRef] [PubMed]

*l*

_{2}norm. So it’s possible to improve the reconstruction results through modifying our

*hp*-FEM algorithm.

*hp-*FEM based algorithm for BLT reconstruction. Both the numerical simulation and physical experiment show that the adaptive

*h*-refinement and

*p*-refinement can achieve a better reconstructed result. One of our major goals is to realize

*hp*-FEM in 3D complex solution region like small animal, and study higher polynomial degree of elements. Furthermore, in order to implement dual source reconstruction more accurately, we can improve the optimization objective function

## Acknowledgments

## References and links

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

2. | C. H. Contag and M. H. Bachmann, “Advances in in vivo 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. | T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. |

5. | D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. |

6. | R. Weissleder, “Scaling down imaging: molecular mapping of cancer in mice,” Nat. Rev. Cancer |

7. | J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. |

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

9. | W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express |

10. | M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. |

11. | 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 |

12. | J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express |

13. | M. Ainsworth and B. Senior, “Aspects of an adaptive |

14. | M. Ainsworth, “A preconditioner based on domain decomposition for |

15. | 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. |

16. | MOSE, http://www.mosetm.net/ |

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

18. | J. Welch, and M. J. C. van Gemert, Optical and Thermal response of laser-irradiated tissue (Plenum Press, New York, 1995). |

19. | T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. |

20. | M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. |

21. | 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. | J. J. Duderstadt, and L. J. Hamilton, Nuclear Reactor analysis (Wiley, New York, 1976). |

23. | S. S. Rao, The finite element method in engineering, (Butterworth-Heinemann, Boston, 1999). |

24. | P. E. Gill, W. Murray, and M. Wright, Practical optimization, (Academic Press, New York, 1981). |

25. | Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. |

26. | D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE |

27. | I. Babuška and B. Guo, “Approximation properties of the |

28. | C. Schwab, “ |

29. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: May 26, 2009

Revised Manuscript: July 17, 2009

Manuscript Accepted: July 22, 2009

Published: August 3, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Runqiang Han, Jimin Liang, Xiaochao Qu, Yanbin Hou, Nunu Ren, Jingjing Mao, and Jie Tian, "A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography," Opt. Express **17**, 14481-14494 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14481

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### References

- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005). [CrossRef] [PubMed]
- C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002). [CrossRef] [PubMed]
- 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).
- T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. 17(5), 545–580 (2003). [CrossRef] [PubMed]
- D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. 6(4), 319–331 (2004). [CrossRef] [PubMed]
- R. Weissleder, “Scaling down imaging: molecular mapping of cancer in mice,” Nat. Rev. Cancer 2(1), 11–18 (2002). [CrossRef] [PubMed]
- J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008). [CrossRef] [PubMed]
- G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004). [CrossRef] [PubMed]
- W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6756 . [CrossRef] [PubMed]
- M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50(12), 2837–2858 (2005). [CrossRef] [PubMed]
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