## Bioluminescence tomography with Gaussian prior |

Biomedical Optics Express, Vol. 1, Issue 5, pp. 1259-1277 (2010)

http://dx.doi.org/10.1364/BOE.1.001259

Acrobat PDF (1406 KB)

### Abstract

Parameterizing the bioluminescent source globally in Gaussians provides several advantages over voxel representation in bioluminescence tomography. It is mathematically unique to recover Gaussians [Med. Phys. **31**(8), 2289 (2004)] and practically sufficient to approximate various shapes by Gaussians in diffusive medium. The computational burden is significantly reduced since much fewer unknowns are required. Besides, there are physiological evidences that the source can be modeled by Gaussians. The simulations show that the proposed model and algorithm significantly improves accuracy and stability in the presence of Gaussian or non- Gaussian sources, noisy data or the optical background mismatch. It is also validated through *in vivo* experimental data.

© 2010 OSA

## 1. Introduction

1. 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. Imaging **16**(4), 378–387 (2002). [PubMed]

3. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [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**(17), 7801–7809 (2006). [PubMed]

13. Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging **2007**, 86741 (2007). [PubMed]

*τ*} of the domain

_{i}, i≤N*Ω*, the source in piecewise constants iswhere

*1*is the basis function supported on the subdomain

_{i}*τ*that is 1 on

_{i}*τ*, and 0 otherwise, and

_{i}*q*is the average intensity value on

_{i}*τ*to be recovered.

_{i}3. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [PubMed]

14. H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express **18**(3), 1854–1871 (2010). [PubMed]

15. H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express **18**(3), 2894–2912 (2010). [PubMed]

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

## 2. Methods

### 2.1. Forward Modeling

21. H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. **55**(4), 947–962 (2010). [PubMed]

22. W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(5 Pt 1), 051913 (2007). [PubMed]

*f*for the light intensity,

*q*for the bioluminescent source,

*μ*for the absorption coefficient,

_{a}*μ*for the reduced scattering coefficient,

_{s}^{'}*D*for the diffusion coefficient, i.e.,

*D*= 1/ [3(

*μ*+

_{a}*μ*)],

_{s}^{'}*A*for the boundary constant [24].

*Φ*, the discretized source

*Q*, and the system matrix

*F*dependent on

*μ*and

_{a}*D*, i.e.,

### 2.2. Source Representation by Gaussians

**31**(8), 2289–2299 (2004). [PubMed]

*a priori*knowledge on the source distribution; in particular the uniqueness can be established when source is assumed to be composed of solid balls with the known intensities. However, using solid balls is not effective in representing general source distribution, especially features like anisotropy and fast decay of source intensity which are quite common in practice. In this study, we use summation of Gaussians to model general source distribution. This structural representation is flexible and improves the reconstruction stability for BLT. In particular the anisotropic Gaussian representation is an effective mathematical approximation in the sense that it parameterizes the major quantitative information of interest, such as the intensity, the center and the size of the source.

*q*as the summation of Gaussians with the intensity

*ρ*, the center (

*x*), anisotropic radius (

_{c}, y_{c}, z_{c}*r*) and Euler angles (j, f, q), i.e.,where

_{x}, r_{y}, r_{z}### 2.3. Gaussian-based BLT

*X*= {

^{G}*ρ*j

_{j}, x_{c,j}, y_{c,j}, z_{c,j}, r_{x,j}, r_{y,j}, r_{z,j},*, f*

_{j}*, q*

_{j}*},*

_{j}, j≤n*f*} for the measured data, the set {

_{i}, i≤M*P*} of column vectors for the discretized measuring operator, and geometric constraints

_{i}, i≤M*R*on

*X*, which can be regarded as the regularization term. Please note that

^{G}*f*is now nonlinearly dependent on

*X*in this new formulation, although it linearly depends on

^{G}*q*in Eq. (1) in the voxel-based representation.

*n*as a variable, mainly because there is generally no practical need for that since the tumors are usually localized and limited in quantity. We can just assign a practical estimate to

*n*that is larger enough than or comparable with the actual number.

**Algorithm 1**: Combinatorial optimization**Given**: initial guess*n*and_{0}*e*= 0.05.- 2.
*d = ||f*_{k}*(**X*_{k}*)*–*f||/||f||;* - 3. Stopping criterion:
**Quit**if*d <e.* - 4.
*n*_{k + 1}= n_{k}+ 1

*e = 0.05*is an empirical number that can identify those

*n*’s that are sufficient for the reconstruction;

*f*denotes the measurements with the source

_{k}*X*; || × || can be the simple summation of the absolute values.

_{k}*R*, which can be imposed naturally from the known geometry of the medium. Please note that we do not assume any

*a priori*knowledge on the anatomical structure of the medium, e.g., from other imaging modalities, although they can be easily incorporated into the scheme to improve the reconstruction. All the geometric constraints are either from the shape of the medium or from some commonsense assumptions on the shape of the source to avoid some non-uniqueness in geometric representation.

*X*. For example, we can use the size of the medium as the min-max constraint for the center, and realistic values for anisotropic radius or intensity such as zeros as the lower bounds. That is

^{G}*0°*£q<

*180°*, the representation of ellipsoid is not unique, e.g., by simultaneously exchanging

*r*and

_{x}*r*and considering the supplementary angle of q in 2D. However, this does not affect the reconstruction result.

_{y}*c*,

_{xy}*c*and

_{yz}*c*control the size difference between any two directions for each source.

_{zx}*i*and

*j*,

*c*,

_{x}*c*and

_{y}*c*are the control parameters, which can for example be set to

_{z}*ln2*, corresponding to full width at half maximum (FWHM). Without using constraints from Eq. (7.2), multiple inclusions may not be separated. An example is given in the result section.

### 2.4. Minimization by Barrier Method

*dX*iteratively via the following outer loop of iterations

*f*is the simulated data with

^{n}*X*,

^{n}*J*is the Jacobian coming from the linearization, which depends on

*X*, and the detail for the computation of

^{n}*J*is given in the next section for the algorithm implementation.

*b = f-f*and

^{n}*x = dX*

*R*. That is we solve a sequence of the following minimization problems

*R*since the value of logarithmic penalty functions would otherwise become infinitely large. During each step the solution

*x*is no more than

^{n}*K/t*-suboptimal with

*K*as the total number of constraints. This implies

*x*converges to the optimal point of Eq. (11) such that

^{n}*Jx = b*with strictly enforced geometry constraints as

*t*→∞. The Eq. (12) can be minimized with Newton’s method via the following inner loop

*dx*, find the moving step

*s*through the backtracking line search, and then update

*x*and

*t*for the next iteration until the stopping criterion is satisfied. That is we solve a sequence of

*t*-subproblems via Eq. (13) with the increasing

*t*.

### 2.5. Algorithm Implementation

**Outer loop**: Linearization of the data fidelity via Eq. (10)**Given**: initial guess*X*and^{0}*e*= 0.01._{o}**Repeat**: 1. Compute Jacobian*J*from*X*.^{n}**Given**:*t*,^{0}= −2R(X^{n})/||b||^{2}*μ = 2*, initial guess*x*.^{0}= 0**Repeat**: 2.1. Compute the descent direction*dx*;- 2.2. Compute the moving step
*s*via backtracking line search; **Given**:*a = 0.01, s = 1, b = 0.5*.**While**:*L(x*^{n}+ sdx)>L(x^{n}) + α(ÑL_{x})^{T}× (sdx)*s =*b*s*.- 2.3. Update
*x*+^{n + 1}= x^{n}*sdx*and*t*.^{n + 1}= μt^{n} - 2.4. Stopping criterion:
**Quit**if*K/t*.^{n}<e_{i} - 3. Stopping criterion:
**Quit**if*|E*^{n + 1}-E^{n}|/ E^{n}<e_{o}.

*E = òqdΩ*as the stopping criterion for the outer loop, i.e., assuming that the total power

*E*is stable when the algorithm converges. As the stopping criterion for the inner loop, we use the sub-optimal ratio

*K/t*, which naturally measures the difference between the iterative solution of Eq. (12) and the original solution with strictly enforced constraints. We set

*e*empirically to

_{i}*0.0001t*. Please note that the formula for

^{0}*t*is for balancing the linearized data fidelity and the penalty function.

^{0}*J = {J*, which can be derived from the Jacobian

_{ij}, i£M, j£n}*J*with respect to the piecewise-constant voxel representation via Eq. (1). The connection between

_{0}= {J_{0,ij}, i£M, j£N}*J*and

*J*is through the following coordinate transformationwhere

_{0}*q*represents the

_{k}*k*th voxel value in Eq. (1),

*x*is the

_{j}*j*th parameter in global representation by Gaussians in Eq. (5).

*J*is through the adjoint method [24]. After computing

_{0}*J*,

_{0}*J*follows immediately from (14). However, noticing that there are only a few parameters in the new source representation by Gaussians, i.e.,

*n†M†N*, we directly compute

*J*without computing

*J*via the direct method as follow.

_{0}*F(∂ϕ/∂q*, then

_{k}) = ∂Q/∂q_{k}*n*times, which would be at least

*M*times if we compute

*J*first via Eq. (14). Please note that

_{0}*∂Q/∂q*is a sparse vector, which is nonzero only at the nodes in the

_{k}*k*th voxel; on the other hand

*∂q*can be analytically derived from Eq. (5). Therefore, we can now compute the Jacobian

_{k}/∂x_{j}*J*merely by computing

*n*forward solvers, which is much more efficient than that in the voxel-based BLT. The trade-off is that we need to compute the Jacobian more than once due to the nonlinear dependence of

*J*on

*X*, which needs to be computed only once in the voxel-based BLT due to the linear dependence of

^{G}*J*on

_{0}*X*. However, since

*n†M*, we still achieve a considerable gain in speed, since the new algorithm usually converges in less than 20 iterations while the ratio

*M/n*is usually at least about a hundred. That is we usually achieve at least 5 gains in speed on the computation of Jacobians.

*n>M*, the adjoint method is preferred, i.e.,That is we compute

*M*forward solvers instead.

*g*as in Eq. (7), i.e.,

_{k}(X^{G})*R*. From the straightforward computation, we have the following simple formula

_{k}= -ln[-g_{k}(X^{G})]*R*is evaluated at

_{k}*X*rather than

^{n}+ x*x*.

*J*in Step 1, although it takes less computational time than that in voxel-based BLT, which is again due to the reduced number of unknowns.

## 3. Results

### 3.1. Reconstruction with Single Inclusion

*in vivo*data.

*μ*and

_{a}= 0.01mm^{−1}*μ*. Without further mentioning, we use

_{s}^{'}= 1mm^{−1}*mm*as the unit for the length,

*mm*as the unit for absorption or scattering coefficients, nano Watts (

^{−1}*nW*) as the unit for the total source power, and

*nW/mm*,

^{2}*nW/mm*as the unit for the source intensity in 2D and 3D respectively. The unit of angle is in degree.

^{3}*ρ = 0.1*,

*x*,

_{c}= 0*y*,

_{c}= 0*r*. The reconstruction with multiple inclusions as the initial guess will be considered in the next section. Please notice that the nonzero value has to be assigned for

_{x}= 1, r_{y}= 1 and q = 90°*ρ*, otherwise the Jacobian with respect to other parameters will be zero according to Eq. (14). For the constraint variables, we set

*ρ*,

_{max}= 10, ρ_{min}= 0.01, x_{max}= y_{max}= 10, x_{min}= y_{min}= −10, r_{x,max}= r_{y,max}= 5, r_{x,min}= r_{y,min}= 0.1*q*, and

_{max}= 180°,q_{min}= 0°*c*.

_{xy}= 5_{2}regularization [24]. That is to minimizewhere the Jacobian

*J*is with respect to the piecewise-constant voxel representation by Eq. (1), and

_{0}*l*is the regularization parameter.

14. H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express **18**(3), 1854–1871 (2010). [PubMed]

15. H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express **18**(3), 2894–2912 (2010). [PubMed]

#### 3.1.1. Inclusions with Gaussian Shapes

*E*and maximal intensity as documented in Table 1.

#### 3.1.2. Inclusions with Various Shapes

*E*and maximal intensity as documented in Table 3.

*r*and

_{x}= 1.752*r*, while the true

_{y}= 1.797*r*and

_{x}= 1*r*. Although

_{y}= 2*r*in the reconstruction, this scale difference is much smaller than the true difference.

_{x}<r_{y}*q*as reconstruction variable. The reason is that G-BLT does not recover the shapes for non-Gaussian sources as exactly as for Gaussian sources even with rotation.

#### 3.1.3. Different Noise Level

*f*respectively, e.g.,

*f(1 + 5% × Randn)*, where

*Randn*represents the random Gaussian distribution with zero mean and unit variance. The reconstructed results with both G-BLT and V-BLT are plotted in Fig. 4 and the parameter values are in Table 4 . Here the simulations are performed on the same circular phantom with single circular inclusion as in Fig. 2(a).

#### 3.1.4. Mismatch of Optical Background

*μ*and

_{a}(1 + 10%)*μ*. Notice that we either increase or decrease both absorption and scattering coefficients since empirically we find that the mismatch may cancel if one is increased while the other is decreased. Therefore, we change both in the same direction to simulate the worst-case errors. Please also note that the reconstruction results with V-BLT are not presented since it does not localize the inclusion satisfactorily even in the case without optical background error.

_{s}^{'}(1 + 10%)### 3.2. Reconstruction with Multiple Inclusions

*n*to a reasonable larger value than the actual number, in order to faithfully reconstruct the source distribution without considering

*n*as an independent variable. Particularly, as the initial guess, we assume there are four Gaussian inclusions, i.e.,

*n = 4*, with

*ρ*,

_{1}= ρ_{2}= ρ_{3}= ρ_{4}= 0.1*x*,

_{1,c}= 5*y*,

_{1,c}= 5*x*,

_{2,c}= 5*y*,

_{2,c}= −5*x*,

_{3,c}= −5*y*,

_{3,c}= −5*x*,

_{4,c}= −5*y*,

_{4,c}= 5*r*,

_{1,x}= r_{2,x}= r_{3,x}= r_{4,x}= 1*r*and

_{1,y}= r_{2,y}= r_{3,y}= r_{4,y}= 1*q*. Notice that for the same reason mentioned above, we do not consider

_{1}= q_{2}= q_{3}= q_{4}= 90°*q*as reconstruction variable for non-Gaussian sources for simplicity.

*c*=

_{x}*c*=

_{y}*ln2*, corresponding to that the minimal distance between any two objects is at least the average of FWHM.

#### 3.2.1. Inclusions with Gaussian Shapes

#### 3.2.2. Inclusions with Non-Gaussian Shapes

#### 3.2.3. Combinatorial optimization

*n*as a variable.

_{0}= 1, and continue the combinatorial process until the data discrepancy

*d*reaches the tolerance. Please notice that the process should have terminated at

*n = 3*. For illustration purpose, we run until

*n = 5*.

*n = 1*and

*n = 2*fail to give a satisfactory reconstruction while those from

*n = 3*,

*n = 4*and

*n = 5*all successfully recover the desired distribution.

*n = 4*and two recovered sources in Fig. 8(f) constitute the left Gaussian in Fig. 8(a) when using

*n = 5.*

### 3.3. 3D in vivo validation

*in vivo*experimental data. The details of experimental setups and mouse experiments are given in [4

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**(17), 7801–7809 (2006). [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**(17), 7801–7809 (2006). [PubMed]

^{3}(right) and a smaller one of the power 1.5nW/mm

^{3}(left). When the mouse was dissected after the experiment, two tumors were found on both adrenal glands, respectively, as shown in Fig. 9(c) . The volume of tumor tissues as measured by Vernier calipers was 468 mm

^{3}for the tumor (right) 275 mm

^{3}for the tumor (left).

*n = 4*, with

*ρ*,

_{1}= ρ_{2}= ρ_{3}= ρ_{4}= 10*x*,

_{1,c}= x_{2,c}= x_{3,c}= x_{4,c}= 0*y*,

_{1,c}= y_{2,c}= y_{3,c}= y_{4,c}= 0*z*,

_{1,c}= 5*z*,

_{2,c}= 10*z*,

_{3,c}= 15*z*,

_{4,c}= 20*r*,

_{1,x}= r_{2,x}= r_{3,x}= r_{4,x}= 1*r*and

_{1,y}= r_{2,y}= r_{3,y}= r_{4,y}= 1*r*.

_{1,z}= r_{2,z}= r_{3,z}= r_{4,z}= 1*ρ*, and

_{max}= 1000, ρ_{min}= 0.1, x_{max}= y_{max}= 12, x_{min}= y_{min}= −12, z_{max}= 27, z_{min}= 0, r_{x,max}= r_{y,max}= 6, r_{x,min}= r_{y,min}= 0.5*c*=

_{xy}= c_{yz}= c_{zx}= 5, c_{x}*c*=

_{y}*c*=

_{z}*ln2*. Here the min-max values on the geometry correspond to the physical size of the domain; the maximum of the intensity is estimated from the data, which is roughly 50 times of the maximal value in the data.

^{3}(right) and the other one with the peak intensity of 7.692nW/mm

^{3}(left). In addition, the peak intensity of the other 2 inclusions (Inclusion 3 and 4) is less than 5% of the maximum (Inclusion 1), which can be from experimental or numerical noise. The reconstructed bioluminescent volumes (

*2r*) are 506 mm

_{x}× 2r_{y}× 2r_{y}^{3}(right) and 901 mm

^{3}(left) respectively. The discrepancy between the reconstructed volumes and the measured ones is not surprising since the anatomical volume may not correspond to the bioluminescent volume.

## 4. Conclusions and discussions

*in vivo*experimental data. In particular, when the source itself can be approximated by Gaussians, the proposed method is able to accurately recover the intensity, centers, radiuses and rotation angles of the Gaussians.

*in vivo*bioluminescence imaging method, for which we may need to incorporate RTE for more accurate modeling, simplified representation for optical heterogeneity, such as piecewise constants, to be simultaneously reconstructed for correcting optical background, and multi-spectral data for better stability.

## Acknowledgments

## References and links

1. | 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. Imaging |

2. | G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology |

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

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. | 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 |

6. | G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. |

7. | G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging |

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

9. | 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. |

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

11. | C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging |

12. | 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. |

13. | Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging |

14. | H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express |

15. | H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express |

16. | J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. |

17. | K. M. Case, and P. F. P. F. Zweifel, |

18. | A. Ishimaru, |

19. | E. E. Lewis, and W. F. Miller, |

20. | H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. |

21. | H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. |

22. | W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

23. | A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. |

24. | A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

25. | S. Boyd, and L. Vandenberghe, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.6960) Imaging systems : Tomography

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: September 20, 2010

Revised Manuscript: October 16, 2010

Manuscript Accepted: October 27, 2010

Published: October 29, 2010

**Citation**

Hao Gao, Hongkai Zhao, Wenxiang Cong, and Ge Wang, "Bioluminescence tomography with Gaussian prior," Biomed. Opt. Express **1**, 1259-1277 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-5-1259

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### References

- 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. Imaging 16(4), 378–387 (2002). [PubMed]
- G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).
- G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004). [PubMed]
- 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(17), 7801–7809 (2006). [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). [PubMed]
- G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
- G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).
- X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004). [PubMed]
- 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(23), 5421–5441 (2005). [PubMed]
- 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(3), 365–367 (2006). [PubMed]
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