## Spatially varying regularization based on spectrally resolved fluorescence emission in fluorescence molecular tomography

Optics Express, Vol. 15, Issue 21, pp. 13574-13584 (2007)

http://dx.doi.org/10.1364/OE.15.013574

Acrobat PDF (158 KB)

### Abstract

Fluorescence molecular tomography suffers from being mathematically ill-conditioned resulting in non-unique solutions to the reconstruction problem. In an attempt to reduce the number of possible solutions in the underdetermined system of equations in the reconstruction, we present a method to retrieve a spatially varying regularization map outlining the feasible inclusion position. This approach can be made very simple by including a few multispectral recordings from only one source position. The results retrieved through tissue phantom experiments imply that initial reconstructions with spatially varying priors reduces artifacts and show slightly more accurate reconstruction results compared to reconstructions using no priors.

© 2007 Optical Society of America

## 1. Introduction

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

*a priori*information [2

2. B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, “Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure,” J. Biomed. Opt. **10** (2005). [CrossRef] [PubMed]

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

4. H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, “Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies,” Appl. Opt. **44**, 2177–2188 (2005). [CrossRef] [PubMed]

5. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

6. R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental Fluorescence Tomography of Tissues With Noncontact Measurements,” IEEE Trans. Med. Imaging **23**, 492–500 (2004). [CrossRef] [PubMed]

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

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

## 2. Theory

### 2.1. Forward light propagation model

_{s}inside the medium. The extrapolated boundary condition is applied using mirrored sources to account for the refractive index mismatch at the boundaries [8

8. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation . 1. Theory,” Appl. Opt. **36**, 4587–4599 (1997). [CrossRef] [PubMed]

*u*[

_{x}*W*/

*m*

^{2}]) at an arbitrary position

**r**due to a point source at position

**r**

_{s}is then analytically given by the Green’s solution to the homogeneous diffusion equation, Eq. (1).

*x*denotes excitation wavelength.

*P*

_{0}[

*W*] is the laser source power,

*D*[

_{x}*m*] and μ

_{effx}[

*m*

^{-1}] is the diffusion coefficient and effective attenuation coefficient, respectively, defined in Eq. (3) and Eq. (4).

_{a}[

*m*

^{-1}] is the absorption coefficient and μ′

_{s}[

*m*

^{-1}] is the reduced scattering coefficient. As before the subscript x denotes excitation while subscript

*m*denotes emission wavelengths. The dimensionless constant α (here α ≈ 0.55) is adopted from Ripoll

*et. al*. [9

9. J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, “Experimental determination of photon propagation in highly absorbing and scattering media,” J. Opt. Soc. Am. **22**, 546–551 (2005). [CrossRef]

**r**the excitation light will be absorbed and the fluorophore will emit fluorescence. The strength of the emitted fluorescence is dependent on the absorption coefficient and the fluorescent yield of the fluorophore. The fluorescent yield is characterized by the broad fluorescence spectrum and provides information about the probability that an excitation photon will induce a fluorescent photon at a specific spectral band.

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

*u*[

_{m}*W*/

*m*

^{2}]) at a detector position

**r**

_{d}due to an excitation event in a small volume

*dV*at a position

**r**is then analytically described by Eq. (5) [11

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

*m*denotes the emission wavelength. μ

_{af}(

**r**) [

*m*

^{-1}] is the fluorophore absorption coefficient at the excitation wavelength at an arbitrary position

**r**. γ

_{m}[–] is the fluorescence yield at emission wavelength λ

_{m}. Considering an arbitrary spatial distribution of fluorophore the fluorescence fluence rate at a detector position

**r**

_{d}, due to an excitation source in

**r**

_{s}, is a contribution of all fluorescent volume fractions

*dV*throughout the volume. Hence the fluorescence fluence rate is given by an integral, Eq. (6) [11

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

### 2.2. Reconstruction of fluorophore inclusion

_{af}, the normalized Born approach is applied [12

12. 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]

**r**

_{d}with the excitation source placed at

**r**

_{s}is given by [12

12. 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]

_{af}(

**r**) the volume integral in Eq. (7) is discretized into

*N*

_{voxels}where voxel

*j*has a volume of ∆

*V*[

_{j}*m*

^{3}].

*i*denotes the source-detector pair. Equation (8) is a set of linear equations that can be expressed in matrix form, i.e.

**U**

_{nb}=

**WX**. The elements of the weight matrix

**W**is stated in Eq. (9), while

**X**.

**U**

_{nb}is a column vector with the number of elements equal to the number of source-dector pairs [12

12. 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]

**X**, the weight matrix

**W**should be inverted. Using Tikhonov regularization [13

13. A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” Siam Review **40**, 636–666 (1998). [CrossRef]

*β*is a regularization parameter that reduces the ill-condition of the inverse problem. The reconstructed image will depend on the regularization parameter since choosing a too high value of

*β*will produce an image with low contrast and low resolution. On the other hand a too small value of

*β*introduces noise [5

5. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

14. H. Dehghani, D. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiol. Measurement **20**, 87–102 (1999). [CrossRef]

2. B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, “Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure,” J. Biomed. Opt. **10** (2005). [CrossRef] [PubMed]

### 2.3. Spatial priors based on fluorescence emission

*β*can be spatially varying since the matrix

*β*

**I**is of the size

*N*

_{voxels}×

*N*

_{voxels}. Hence each voxel can have a voxel-specific

*β*-value. The purpose of using a spatially varying regularization parameter is to increase the spatial resolution in regions where the inclusion most likely is positioned. At the same time regions where the inclusion is not positioned should be smoothed to reduce artifacts. To retrieve the spatially varying regulariza-tion parameter, or regularization map, the fluorescence emission from the inclusion itself can be used.

_{m1}and λ

_{m2}, from a fluorescent inclusion detected at the boundary of the tissue is dependent on the depth of the inclusion [15

15. J. Swartling, J. Svensson, D. Bengtsson, K. Terike, and S. Andersson-Engels, “Fluorescence spectra provide information on the depth of fluorescent lesions in tissue,” Appl. Opt. **44**, 1934–1941 (2005). [CrossRef] [PubMed]

16. J. Svensson and S. Andersson-Engels, “Modeling of spectral changes for depth localization of fluorescent inclusion,” Opt. Express **13**, 4263–4274 (2005). [CrossRef] [PubMed]

_{am1}= μ

_{am2}. Following the formalism above, the ratio of the two fluorescence signals induced at a location

**r**is given by

**26**, 893–895 (2001). [CrossRef]

17. S. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, R41–R93 (1999). [CrossRef]

18. J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, “Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues,” J. Opt. Soc.Am. **20**, 714–727 (2003). [CrossRef]

*G*(

_{m}**r**,

**r**

_{d}) =

*G*(

_{m}**r**

*,*

_{d}**r**). Calculation of Eq. (12) from detector

*d*yields a map of the ratio throughout the geometry. For a given set of optical properties the value of the ratio, in Eq. (12) is only dependent on the distance from the detector. Due to the spherical symmetry of the Green’s solution all voxels positioned at the same distance from the detector will have the same value of the calculated ratio. Comparing the forward modelled ratio in each voxel and the experimentally acquired ratio, i.e.

*Ũ*(

_{mR}**r**

_{d}), will then result in a minima in all voxels where the concurrence between the modelled ratio and the measured ratio is optimal. The comparison is performed by simply taking the difference between the two quantities which can be expressed as

*j*denotes voxel number in the discretized geometry and

*Ũ*(

_{mR}**r**

_{d}) represents the ratio of the measured fluorescence intensity at the two wavelengths. In the evaluation of the difference between modelled and measured values, we have in Eq. (13

13. A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” Siam Review **40**, 636–666 (1998). [CrossRef]

13. A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” Siam Review **40**, 636–666 (1998). [CrossRef]

**26**, 893–895 (2001). [CrossRef]

**40**, 636–666 (1998). [CrossRef]

*p*is the matrix element of voxel

_{j,j}*j*in the diagonal matrix

**P**. The incorporation of the regularization map

**P**into the reconstruction algorithm is governed by replacing the matrix

*β*

**I**in Eq. (11) with the matrix

*β*

**P**. The regularization parameter will then have a minimum value of

*β*which is iteratively decreased during the reconstruction. The initial value of the parameter is ten times the maximum diagonal element of the matrix

**W**

^{T}

**W**.β is decreased until the projection error, i.e. |

*Ũ*-

_{nb}**WX**| /

*N*

_{voxels}, is lower than 0.05.

## 3. Experimental setup

### 3.1. Imaging system

*μ*m, the power at the distal end of the fiber was approximately 80 mW. This end of the fiber was translated over one of the boundaries in a source grid made of black delrin plastic with drilled holes providing a source position separation of 5 mm. Emitted fluorescence from the object was detected by a multispectral imaging system consisting of a CCD-camera (C4742-80-12AG, Hamamatsu), a liquid crystal tunable filter (LCTF VIS 20–35, Varispec) and a camera objective lens (Nikon f/1.8, focal length 50 mm). The LCTF filter was scanned in the wavelength range of 532–660 nm in steps of 10 nm to collect light. Each spectral band has a spectral width of 20 nm.

### 3.2. Tissue phantom

19. J. Dam, T. Dalgaard, P. Fabricius, and S. Andersson-Engels, “Multiple polynomial regression method for determination of biomedical optical properties from integrating sphere measurements,” Appl. Opt. **39**, 1202–1209 (2000). [CrossRef]

*d*=5.6 mm) was filled with the phantom solution and Rhodamine 6G, a fluorescent dye with similar emission spectra to fluorescent proteins. Two concentrations of Rhodamine 6G were used

_{cyl}*c*

_{1}=0.5

*μM*and

*c*

_{2}=1

*μM*. The emission characteristics of the fluorophore was acquired using a spectrometer (USB2000, OceanOptics) and the normalized spectrum is shown in Fig. 2(c).

### 3.3. Measurement procedure and data analysis

*z*= 7 or

*z*= 11 mm. Images were acquired with the filter centered at the excitation wavelength 532 nm and at the following fluorescence wavelengths 550–660 nm in steps of 10 nm. Background images were acquired with no excitation light. Images with no fluorescent object inserted into the phantom were also obtained to characterize the autofluorescence from the phantom material. For all measurements the exposure times were in the range of 2–6 seconds.

*Q*= 0.95 is the quatum yield for Rhodamine 6G [20

_{e}20. R. Kubin and A. Fletcher, “Fluorescence Quantum Yields of Some Rhodamine Dyes,” J. Lumin. **27**, 455–462 (1982). [CrossRef]

## 4. Results and Discussion

### 4.1. Spatially varying regularization map based on fluorescence emission

*μM*was placed at

*z*= 11 mm. Calculation of Eq. (13) using emission measurements at λ

_{m1}= 560 nm and λ

_{m2}= 600 nm for three detectors yields the cross-sectional images shown in Fig. 3. The measurements were extracted with the source position fixed, indicated in Fig. 3. The resulting arc is centered around the specific detector, since the minimum is obtained for all voxels positioned at the same distance from the detector. The minimum value for the three detectors shown in the figure are placed at different distances from the detectors due to the fact that the measured ratios

*Ũ*are different, requiring a matching value of |

_{mR}**r**

_{j}-

**r**

_{d}| in the forward model

*u*to compensate. A longer propagation distance yields a smaller intensity ratio, hence the arc is placed further away from the detector.

_{mR}14. H. Dehghani, D. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiol. Measurement **20**, 87–102 (1999). [CrossRef]

*μM*concentration placed at

*z*= 11 mm

*z*= 7 mm is shown in Fig. 4(a) and (b), respectively. The regularization maps for the case with 0.5

*μM*concentration placed at

*z*= 11 mm

*z*= 7 mm is shown in Fig. 4(c) and (d), respectively. All regularization maps retreived through the emission ratio within this paper are based on

*λ*

_{m1}= 560 nm and λ

_{m1}= 600 nm. The higher absorption for

*λ*

_{m1}will produce maps with higher spatial resolution as seen in Fig. 4(b) and (d). Inclusions positioned far from the detection boundary will also have a slightly more narrow intensity distribution over the boundary. This is seen in the maps where Fig. 4(b) and (d) are extended in x-direction while Fig. 4(a) and (c) are more symmetric around the inclusion. The method utilizes a ratio of two emission spectral bands where the dependence on the fluorophore concentration disappears. This is also seen in Fig. 4(a)–(d) where the regularization parameter values are effectively the same. Small differences are due to experimental fluctuations.

### 4.2. Reconstruction of single inclusion

*μM*positioned at

*z*= 11 mm is shown in Fig. 5(a). The same reconstruction using the spatially varying regu-larization map in Fig 4(c) is shown in Fig. 5(b). The cross-sectional plots in z- and x-direction is inserted in Fig. 5(c) and (d). The target fluorophore absorption coefficient was μ

_{af}= 23

*m*

^{-1}indicated in Fig. 5(c) and (d). It is seen that the prior regularization map reduces the artifacts around the inclusion as well as around source and detector boundaries. This effect is due to the higher value of the regularization parameter further away from the inclusion which effectively smoothes the solution.

5. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

### 4.3. Reconstruction of two inclusions

*μM*were placed at a depth of

*z*= 9 mm and imaged using the imaging system. To render the regularization map several source positions were used. Equation (15) was then calculated twice; one with the fluorescence measurements acquired using source positions in the interval

*x*∈ [0,25] yielding

_{source}**P**

_{1}and the other with

*x*∈ [30,50] resulting in

_{source}**P**

_{2}. The selection of source positions are chosen based on the intensity distribution of the fluorescence emission at the detector side. Hence the regularization map

**P**

_{1}is based on only those measurement where one of the fluorophores is excited while the excitation of the other inclsuion result in

**P**

_{2}. The two regularization maps are then added together by selecting the minimum value of either

**P**

_{1}or

**P**

_{2}for a specific voxel. The resulting regularization map is seen in Fig. 6(b). The reconstructions were performed as above and the result without prior information is seen in Fig. 6(a) and with prior information in Fig. 6(c). As before the artifacts are reduced when reconstructing with prior information. Quantitatively the prior has less impact. This is due to the small dynamics of the scaling ranging only from ~ 1.5 to ~ 3.5, see Fig. 6(b). The reconstructed values of the two inclusions are in both cases, i.e. with and without prior, about 50 % of the true value.

## 5. Conclusion

## Acknowledgements

## References and links

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

2. | B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, “Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure,” J. Biomed. Opt. |

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

4. | H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, “Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies,” Appl. Opt. |

5. | B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

6. | R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental Fluorescence Tomography of Tissues With Noncontact Measurements,” IEEE Trans. Med. Imaging |

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

8. | D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation . 1. Theory,” Appl. Opt. |

9. | J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, “Experimental determination of photon propagation in highly absorbing and scattering media,” J. Opt. Soc. Am. |

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

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

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

13. | A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” Siam Review |

14. | H. Dehghani, D. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiol. Measurement |

15. | J. Swartling, J. Svensson, D. Bengtsson, K. Terike, and S. Andersson-Engels, “Fluorescence spectra provide information on the depth of fluorescent lesions in tissue,” Appl. Opt. |

16. | J. Svensson and S. Andersson-Engels, “Modeling of spectral changes for depth localization of fluorescent inclusion,” Opt. Express |

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

18. | J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, “Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues,” J. Opt. Soc.Am. |

19. | J. Dam, T. Dalgaard, P. Fabricius, and S. Andersson-Engels, “Multiple polynomial regression method for determination of biomedical optical properties from integrating sphere measurements,” Appl. Opt. |

20. | R. Kubin and A. Fletcher, “Fluorescence Quantum Yields of Some Rhodamine Dyes,” J. Lumin. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(260.2510) Physical optics : Fluorescence

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: August 8, 2007

Revised Manuscript: September 25, 2007

Manuscript Accepted: September 26, 2007

Published: October 1, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Johan Axelsson, Jenny Svensson, and Stefan Andersson-Engels, "Spatially varying regularization based on spectrally resolved fluorescence emission in fluorescence molecular tomography," Opt. Express **15**, 13574-13584 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13574

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### References

- V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005). [CrossRef] [PubMed]
- B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005). [CrossRef] [PubMed]
- M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005). [CrossRef] [PubMed]
- H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, "Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies," Appl. Opt. 44, 2177-2188 (2005). [CrossRef] [PubMed]
- B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999). [CrossRef]
- R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004). [CrossRef] [PubMed]
- E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004). [CrossRef]
- D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. 1. Theory," Appl. Opt. 36, 4587-4599 (1997). [CrossRef] [PubMed]
- J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005). [CrossRef]
- D. Paithankar, A. Chen, B. Pogue, M. Patterson, and E. Sevick-Muraca, "Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media," Appl. Opt. 36, 2260-2272 (1997). [CrossRef] [PubMed]
- M. O’Leary, D. Boas, X. Li, B. Chance, and A. 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]
- A. Neumaier, "Solving ill-conditioned and singular linear systems: A tutorial on regularization," Siam Review 40, 636-666 (1998). [CrossRef]
- H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999). [CrossRef]
- J. Swartling, J. Svensson, D. Bengtsson, K. Terike, and S. Andersson-Engels, "Fluorescence spectra provide information on the depth of fluorescent lesions in tissue," Appl. Opt. 44, 1934-1941 (2005). [CrossRef] [PubMed]
- J. Svensson and S. Andersson-Engels, "Modeling of spectral changes for depth localization of fluorescent inclusion," Opt. Express 13, 4263-4274 (2005). [CrossRef] [PubMed]
- S. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999). [CrossRef]
- J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003). [CrossRef]
- J. Dam, T. Dalgaard, P. Fabricius, and S. Andersson-Engels, "Multiple polynomial regression method for determination of biomedical optical properties from integrating sphere measurements," Appl. Opt. 39, 1202-1209 (2000). [CrossRef]
- R. Kubin and A. Fletcher, "Fluorescence Quantum Yields of Some Rhodamine Dyes," J. Lumin. 27, 455-462 (1982). [CrossRef]

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