## Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution

Optics Express, Vol. 15, Issue 15, pp. 9722-9730 (2007)

http://dx.doi.org/10.1364/OE.15.009722

Acrobat PDF (222 KB)

### Abstract

Fluorescence molecular tomography (FMT) has become an important method for in-vivo imaging of small animals. It has been widely used for tumor genesis, cancer detection, metastasis, drug discovery, and gene therapy. In this study, an algorithm for FMT is proposed to obtain accurate and fast reconstruction by combining an adaptive mesh refinement technique and an analytical solution of diffusion equation. Numerical studies have been performed on a parallel plate FMT system with matching fluid. The reconstructions obtained show that the algorithm is efficient in computation time, and they also maintain image quality.

© 2007 Optical Society of America

## 1. Introduction

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**, 313–320 (2005). [CrossRef] [PubMed]

2. E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. **4**, 419–430 (2004). [CrossRef] [PubMed]

3. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**, 158–160 (1996). [CrossRef] [PubMed]

5. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. **30**, 901–911 (2003). [CrossRef] [PubMed]

6. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. **36**, 2260–2272 (1997). [CrossRef] [PubMed]

8. W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE **5535**, 679–686 (2004). [CrossRef]

9. M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parameterization and Bayesian conditioning on photon migration measurements,” Appl. Opt. **38**, 2138–2150 (1998). [CrossRef]

10. M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. **99**, 9619–9624 (2002). [CrossRef] [PubMed]

11. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

13. A. Joshi, W. Bangerth, K. Hwang, J. Rasmussen, and E. M. Sevick-Muraca, “Plane wave fluorescence tomography with adaptive finite elements,” Opt. Lett. **31**, 193–195 (2006). [CrossRef] [PubMed]

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

*et al*. [11

11. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

*et al*. [12

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

15. J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E **64**, 051917 (2001). [CrossRef]

16. S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. **34**, 7395–7409 (1995). [CrossRef] [PubMed]

17. R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization **9**, 877–900 (1999). [CrossRef]

## 2. Methods

### 2.1 Forward problem

*U*(

_{m}*r*,

_{s}*r*) detected at position

_{d}*r*by a steady light source at position

_{d}*r*with the emission wavelength

_{s}*λ*can be formulated as [4

_{m}4. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. **26**, 893–895 (2001). [CrossRef]

18. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A **21**, 231–241 (2004). [CrossRef]

*λ*.

_{m}*λ*, and

_{x}*λ*.

_{m}*r*, and

*r*to the detector at location

*r*.

_{d}*S*

_{0}is a calibration factor that accounts for the system error such as laser power and the unknown gain and attenuation factors of the system.

*n*(

*r*) is the fluorescence distribution representing the fluorochrome concentration at location

*r*multiplied by the fluorescent yield. For a single source-detector pair (

*s*

_{1},

*d*

_{1}), Eq. (1) can be discretized in vector form as follows:

*N*is the number of the discretized voxels. For

*M*source-detector pairs, the weight matrix is generated as

*W*is an element of the weight matrix

_{ij}*W*representing the weight of voxel

*j*to measurement

*i*.

### 2.2 Reconstruction method

_{1},Θ

_{2},…,Θ

*} represent a sequence of hexahedron mesh levels of the volume of investigation, in which the mesh changes from coarse to fine with the increase in the mesh level*

_{k}*k*. Therefore, Eq. (3) on each level can be written as

*k*represents the kth level. Practically, weight matrix

*W*at the

_{k}*k*th level is usually ill-posed, because of the inherent property of diffusion equation and the existence of noise. Therefore, it is usually impractical to solve for fluorescence distribution

*n*from the linear system directly. In the algorithm proposed, an optimization approach is employed by minimizing objective function Ψ

_{k}*(*

_{k}*n*) based on the Tikhonov regularization method as follows:

_{k}*U*(

_{m}*r*,

_{s}*r*) is the fluorescence measurements and

_{d}*λ*is the regularization parameter.

_{k}*n*and

^{L}_{k}*n*are the lower and upper bounds of the fluorescence distribution at the

^{U}_{k}*k*th level. Ψ

*(*

_{k}*n*) is optimized using Interior/CG [17

_{k}17. R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization **9**, 877–900 (1999). [CrossRef]

*s*is a vector of slack variables and

*ν*>0. The optimization consists of finding solutions of Eq. (6) for a sequence of positive barrier parameters {

*ν*} that converges to zero. When switching to a finer mesh, the final solution

_{l}*n*of the

^{r}_{k}*k*th level is used to generate the initial value

*n*

^{0}

_{k+1}of the (

*k*+1)th level as follows:

*k*+1)th level is performed near the solution, which improves the reconstruction stability and accelerates the convergence speed.

*posteriori*error estimates play an important role in adaptive mesh refinement. These estimates use the computed fluorescence distribution on a coarse mesh to indicate the cells where mesh refinement will be most beneficial for the reduction of measurement error. In this work, the direct maximum selection method [12

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

### 2.3 Software implementation

*k*=1. Then, on each mesh level, optimization is performed until either (i) the number of major interior/CG iterations exceeds the maximum iteration number

*L*

_{max}, or (ii) the relative change in the parameter estimates between the last two steps is less than the predefined threshold ξ. Mesh refinement is then performed. The process iterates until the mesh level exceeds the maximum mesh level

*K*

_{max}. In this work,

*L*

_{max}=40, ξ=10

^{-7}, and

*K*

_{max}=3 were used.

16. S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. **34**, 7395–7409 (1995). [CrossRef] [PubMed]

*W*on each mesh level. Therefore, the algorithm is suitable for the parallel plate FMT system with matching fluid as described in Grave’s study [5

_{k}5. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. **30**, 901–911 (2003). [CrossRef] [PubMed]

## 3. Computational experiments

*et al*. [5

5. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. **30**, 901–911 (2003). [CrossRef] [PubMed]

*Z*=0, as shown in Fig. 1(b). Detection points were arranged in a space-filling grid pattern with a tip-tip distance of 0.15 cm at the detection plane

*Z*=1.5 cm, as shown in Fig. 1(c). The discretized voxel size was selected as 0.01×0.01×0.01 cm for synthetic measurements. Two percent of random Gaussian noise is then added to these synthetic measurements.

### 3.1 Single fluorescent target

*µ*′

*of 6 cm*

_{s}^{-1}and an absorption coefficient

*µ*of 0.3 cm

_{a}^{-1}for both the excitation and emission wavelengths. The optical properties used were consistent with estimates of the bulk optical properties of mouse tissues [20

20. E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla. , April 7–10, 2002. [PubMed]

**30**, 901–911 (2003). [CrossRef] [PubMed]

*x*=0 cm,

*y*=0 cm,

*z*=0.75 cm), which was at a depth of 0.75 cm from the illumination surface. The size of the cubic fluorescent target was 0.01×0.01×0.01 cm. The fluorescence distribution

*n*was set to unit 1 in the target and unit 0 in the background.

### 3.2 Dual fluorescent targets

## 4. Results

### 4.1 Single fluorescent target

*n*=0 and the upper bound was set to

^{L}_{k}*n*=100. Tomographic reconstructions were obtained as illustrated in Fig. 2(d) with zero initial guess

^{U}_{k}*n*

^{0}

_{1}=

*n*

^{L}_{1}and regulation parameter

*λ*=10

_{k}^{-9}.

*W*with double precision. But memory of 0.079 GB would be needed in a problem with 2888 unknowns. Obviously, adaptive refinement adopted largely decreases the number of unknowns, which thus reduces computational cost and enhances computational robustness.

*x*=-0.02 cm,

*y*=0.01 cm,

*z*=0.75 cm), which is close to the real one. The reconstructed target fluorescence distribution

*n*was 1.1, with relative error (

_{recons}*RE*=|

*n*-

_{real}*n*|/|

_{recons}*n*|) 10% to the real target fluorescence distribution

_{real}*n*. The relative error is reasonable, because fluorescent targets with different fluorescence distribution and different size may have similar measurements, especially in the presence of noise. It could be illustrated by the 2.0% measurement error

_{real}*RE*=‖

_{meas}*U*

_{m,meas}-

*U*

_{m,cal}‖/‖

*U*

_{m,meas}‖ between the measurements

*U*

_{m,meas}and the theoretically calculated data

*U*

_{m.cal}, which is equal to the Gaussian noise percentage added to the measurement data.

**30**, 901–911 (2003). [CrossRef] [PubMed]

21. X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Bio. **47**, N1–N10 (2002). [CrossRef]

15. J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E **64**, 051917 (2001). [CrossRef]

11. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

### 4.2 Dual fluorescent targets

## 5. Discussion and conclusion

*et al*. [5

**30**, 901–911 (2003). [CrossRef] [PubMed]

15. J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E **64**, 051917 (2001). [CrossRef]

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

25. J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. **91**, 103901 (2003). [CrossRef] [PubMed]

## 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. | E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. |

3. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. |

4. | V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. |

5. | E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. |

6. | D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. |

7. | R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express |

8. | W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE |

9. | M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parameterization and Bayesian conditioning on photon migration measurements,” Appl. Opt. |

10. | M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. |

11. | A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express |

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

13. | A. Joshi, W. Bangerth, K. Hwang, J. Rasmussen, and E. M. Sevick-Muraca, “Plane wave fluorescence tomography with adaptive finite elements,” Opt. Lett. |

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

15. | J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E |

16. | S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. |

17. | R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization |

18. | E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A |

19. | W. Bangerth, “Adaptive finite element methods for the identification of distributed parameters in partial differential equations,” Ph.D. thesis (University of Heidelberg, 2002). |

20. | E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla. , April 7–10, 2002. [PubMed] |

21. | X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Bio. |

22. | R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging |

23. | J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. |

24. | J. Ripoll and V. Ntziachristos, “Iterative boundary method for diffuse optical tomography,” J. Opt. Soc. Am. A |

25. | J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. |

**OCIS Codes**

(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:**

Medical Optics and Biotechnology

**History**

Original Manuscript: March 20, 2007

Revised Manuscript: May 30, 2007

Manuscript Accepted: May 30, 2007

Published: July 19, 2007

**Virtual Issues**

Vol. 2, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Daifa Wang, Xiaolei Song, and Jing Bai, "Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution," Opt. Express **15**, 9722-9730 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-15-9722

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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, 313-320 (2005). [CrossRef] [PubMed]
- E. E. Graves, R. Weissleder, and V. Ntziachristos, "Fluorescence molecular imaging of small animal tumor models," Curr. Mol. Med. 4, 419-430 (2004). [CrossRef] [PubMed]
- M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, "Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158-160 (1996). [CrossRef] [PubMed]
- V. Ntziachristos and R. Weissleder, "Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation," Opt. Lett. 26, 893-895 (2001). [CrossRef]
- E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeter resolution fluorescence molecular imaging system for small animal imaging," Med. Phys. 30, 901-911 (2003). [CrossRef] [PubMed]
- D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, "Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media," Appl. Opt. 36, 2260-2272 (1997). [CrossRef] [PubMed]
- R. Roy and E. M. Sevick-Muraca, "Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation," Opt. Express 4, 353-371 (1999). [CrossRef] [PubMed]
- W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," Proc. SPIE 5535, 679-686 (2004). [CrossRef]
- M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, "Biomedical optical tomography using dynamic parameterization and Bayesian conditioning on photon migration measurements," Appl. Opt. 38, 2138-2150 (1998). [CrossRef]
- M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, "Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets," Proc. Nat. Acad. Sci. 99, 9619-9624 (2002). [CrossRef] [PubMed]
- A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, "Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Express 12, 5402-5417 (2004). [CrossRef] [PubMed]
- Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, J. Shi, and H. Li, "A multilevel adaptive finite element algorithm for bioluminescence tomography," Opt. Express 14, 8211-8223 (2006). [CrossRef] [PubMed]
- A. Joshi, W. Bangerth, K. Hwang, J. Rasmussen, and E. M. Sevick-Muraca, "Plane wave fluorescence tomography with adaptive finite elements," Opt. Lett. 31, 193-195 (2006). [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]
- J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 051917 (2001). [CrossRef]
- S. R. Arridge, "Photon-measurement density functions. Part I: Analytical forms," Appl. Opt. 34, 7395-7409 (1995). [CrossRef] [PubMed]
- R. H. Byrd, M. E. Hribar, and J. Nocedal, "An interior point algorithm for large scale nonlinear programming," SIAM J. Optimization 9, 877-900 (1999). [CrossRef]
- E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography," J. Opt. Soc. Am. A 21, 231-241 (2004). [CrossRef]
- W. Bangerth, "Adaptive finite element methods for the identification of distributed parameters in partial differential equations," Ph.D. thesis (University of Heidelberg, 2002).
- E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, "In vivo time-resolved optical spectroscopy of mice," presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7-10, 2002. [PubMed]
- X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, "Projection access order in algebraic reconstruction technique for diffuse optical tomography," Phys. Med. Bio. 47, N1-N10 (2002). [CrossRef]
- R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental fluorescence tomography of tissues with noncontact measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004). [CrossRef] [PubMed]
- J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, "Fast analytical approximation for arbitrary geometries in diffuse optical tomography," Opt. Lett. 27, 527-529 (2002). [CrossRef]
- J. Ripoll and V. Ntziachristos, "Iterative boundary method for diffuse optical tomography," J. Opt. Soc. Am. A 20, 1103-1110 (2003). [CrossRef]
- J. Ripoll, R. B. Schulz, and V. Ntziachristos, "Free-space propagation of diffuse light: theory and experiments," Phys Rev Lett. 91, 103901 (2003). [CrossRef] [PubMed]

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