## Three-dimensional scheme for time-domain fluorescence molecular tomography based on Laplace transforms with noise-robust factors

Optics Express, Vol. 16, Issue 10, pp. 7214-7223 (2008)

http://dx.doi.org/10.1364/OE.16.007214

Acrobat PDF (374 KB)

### Abstract

As a visualizing and quantitative method, Fluorescence Molecular Tomography (FMT) has many potential applications in biomedical field and its three-dimensional (3D) implementation is needed in both theory and practice. In this paper, we propose a 3D scheme for time-domain FMT within the normalized Born-ratio formulation. A finite element method solution to the Laplace transformed time-domain coupled diffusion equations is employed as the forward model, and the resultant linear inversions at two distinct transform-factors are solved with an algebraic reconstruction technique to separate fluorescent yield and lifetime images. The algorithm is validated using simulated data for 3D cylinder phantoms, and the spatial resolution and quantitativeness of the reconstruction assessed. We demonstrate that the proposed approach can accurately retrieve the positions and shapes of the targets with high spatial resolution and quantitative accuracy, and tolerate a signal-to-noise ratio down to 25dB by appropriately choosing the transform factors.

© 2008 Optical Society of America

## 1. Introduction

1. R. Weissleder, C.H. Tung, U. Mahmood, and A. Bogdanov, “In vivo imaging with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. **17**, 375–378 (1999). [CrossRef] [PubMed]

7. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulation,” Appl. Opt. **37**, 5337–5343 (1998). [CrossRef]

*et al*used the CW to image the tumors targeted by special fluorescence molecular probes in mice. The fluorescence lifetime image reconstructed by frequency-domain and time-domain models can provide further information about the surroundings of the biological fluorophore, such as pH, enzyme, oxygen,

*etc.*[12–16

12. B. Yuan and Q. Zhu, “Separately reconstructing the structural and functional parameters of a fluorescent inclusion embedded in a turbid medium,” Opt. Express **14**, 7172–7187 (2006). [CrossRef]

7. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulation,” Appl. Opt. **37**, 5337–5343 (1998). [CrossRef]

*et al*proposed Bayesian image reconstruction for 3D fluorescence lifetime tomography using experimental measurement [8

8. A.B. Milstein, S. Oh, K.J. Webb, C.A. Bouman, Q Zhang, D.A. Boas, and R.P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. **42**, 3081–94 (2003). [CrossRef] [PubMed]

*et al*separated the imaging procedure into two steps to reconstruct the fluorescent yield and lifetime respectively [11–12

11. A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys. **32**, 992–1000 (2005) [CrossRef] [PubMed]

*et al*[13

13. A. B Milstein, J.J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman, and K. J. Webb, “Fluorescence optical diffusion tomography using multiple-frequency data”, Opt. Soc. Am. A **21**,1035–1049 (2004) [CrossRef]

16. F. Gao, H.J. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomogrphy,” Opt. Express **14**, 7109–7124(2006). [CrossRef] [PubMed]

17. V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, A. Sardini, P. M. W. French, and S. R. Arridge “Fluorescence lifetime tomography by using time-gated data acquisition,” Appl. Opt. **46**, 7384–7391(2007). [CrossRef] [PubMed]

## 2. Methodology

### 2.1 Forward model

*x*and

*m*denote the excitation and emission wavelengths respectively;

*c*is the speed of light in medium; the optical properties involved are the absorption coefficient μ

_{aν}(r)(

*ν*∈[

*x*,

*m*]), the reduced scattering coefficient

*μ*′

_{sν}(

**r**)(mm

^{-1}) and the diffusion coefficient

*(*

**κ**_{ν}**r**,

**t**)=c/[3μ′

_{sν}(

**r**)]; the fluorescent yield

*ημ*(

_{af}**r**) (product of the quantum efficiency and absorption coefficient of the fluorophore) and lifetime

*τ*(

**r**) are the fluorescence properties. In Eq. (1), the source term of the excitation equation,

*i.e.*,

*S*(

_{x}**r**,

**r**

_{s},

*t*)=

*δ*(

**r**-

**r**

_{s},

*t*), is an impulse function at excitation position

**r**

_{s}=1/μ′

_{sx}and

*t*=0, while the source term of the emission equation,

*i.e.*,

*s*(

_{m}**r**,

**r**

_{s},

*t*)=

*ημ*(

_{af}**r**)∫

^{t}

_{0}

*c*Φ

_{x}(

**r**,

*t*′)Γ(

**r**)e

^{[-(t-t′)Γ(r)]}d

*t*′(Γ(

**r**)=1/

*τ*(

**r**)), is originated from the propagation of the excitation light in fluorescent target, and might be distributed throughout the medium.

*β*, the following steady-state coupled equations are obtained [16

16. F. Gao, H.J. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomogrphy,” Opt. Express **14**, 7109–7124(2006). [CrossRef] [PubMed]

**Φ**

_{ν}(

**r**,

**r**

_{s},

*β*) is the Laplace transform of the time-dependent photon density Φ

_{ν}(

**r**,

**r**

_{s},

*t*). The measurable flux

*I*

_{ν}(

**r**

*,*

_{d}**r**

*,*

_{s}*β*) at the boundary can be simply calculated by the Fick’s law under Robin boundary condition,

*i.e.*,

*I*(

_{ν}**r**

*,*

_{d}**r**

*,*

_{s}*β*)=

*c*Φ

*(*

_{ν}**r**

*,*

_{d}**r**

*,*

_{s}*β*)(1-

*R*)/[2(1+

_{f}*R*)], where

_{f}*R*≈0.53 represents the internal reflection coefficient at the air-tissue boundary. In this paper, the coupled diffusion equation is solved using a finite element method with the support of the extrapolated boundary condition [16

_{f}16. F. Gao, H.J. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomogrphy,” Opt. Express **14**, 7109–7124(2006). [CrossRef] [PubMed]

### 2.2 Inverse issue

*i.e.*, the inverse issue, is to recover the three-dimensional map of the fluorescent property from the boundary measurements at emission wavelength. In this work the normalized Born-ratio formulation is employed to improve image quality for its advantages of independence of source intensity and less sensitivity to systematic errors [25

25. A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging **24**, 1377–1386 (2005). [CrossRef] [PubMed]

**r**

*with the excitation source at*

_{d}**r**

*, and an integral equation is obtained according to Eq. (2)*

_{s}*x*(

**r**,

*β*)=

*ημ*(

_{af}**r**)/[1+

*βτ*(

**r**)];

*G*(

_{m}**r**

_{d},

**r**,

*β*) is the Green function of emission diffusion equation that describes the propagation of the emission wave emitted from a fluorochrome at

**r**to a detector at

**r**

*, and complies with the reciprocity theorem.*

_{d}*E*elements joined at

*N*vertex nodes by finite element method. Accordingly, the volume integral in Eq. (3) reduces to the following

*V*corresponds to a particular voxel in the discretized geometry.

_{e}*Ḡ*(

_{m}**r**

*,*

_{d}**r**,

*β*),

_{x}(

**r**,

**r**

_{s},

*β*) and

*x̄*(

**r**,

*β*) denote mean values in

*V*. Eq. (4) is a set of linear equations that can be expressed in matrix form

_{e}*I*(

^{nb}*β*)=[

*I*(

^{nb}**r**

^{1}

*,*

_{s}**r**

^{1}

*,*

_{d}*β*),

*I*(

^{nb}**r**

^{2}

*,*

^{s}**r**

^{1}

*,*

_{d}*β*),…,

*I*(

^{nb}**r**

*,*

^{S}_{s}**r**

*,*

^{D}_{d}*β*)]

*,*

^{T}**r**

*and*

^{i}_{s}**r**

*denote the*

^{j}_{d}*i*th source and the

*j*th detector respectively. The unknown quantity

*x*(

**r**,

*β*)≈

**x**

*(*

^{T}*β*)

**u**(

**r**), where

**x**(

*β*)=[

*x*

_{1}(

*β*),

*x*

_{2}(

*β*), …

*x*(

_{N}*β*)]

*and*

^{T}**u**(

**r**)=[

*u*

_{1}(

**r**),

*u*

_{2}(

**r**),…,

*u*(

_{N}**r**)]

*is the shape function.*

^{T}**W**(

*β*)is a matrix of

*S*×

*D*rows and

*N*columns. Any element in the matrix can be given as

*V*numerates all the elements joined at the nth node;

_{e}*G*(

_{m}**r**

*,*

^{i}_{d}**r**,

*β*) and Φ

_{x}(

**r**,

**r**

*,*

^{i}_{s}*β*) at the nodes of the element respectively; Eq. (5) is usually of large scale and ill-posed. The algebraic reconstruction technique (ART) is used to solve the inverse problem. This method generates reconstructions via an iterative process which begins with an initial estimate of the object being reconstructed and then improves on this initial estimate via a sequence of estimates that presumable converge to ‘optimum’ reconstruction after some large number of iterations[25

25. A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging **24**, 1377–1386 (2005). [CrossRef] [PubMed]

*β*possess different capabilities for extracting spatial information. The choice of

*β*value is a trade-off between resolution (better at earlier times) and the signal-to-noise ratio (SNR) (better at later times) [23

23. V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. **32**, 2034–2036 (2007). [CrossRef] [PubMed]

27. J Wu, L. Perelman, R. R. Dasari, and M. S. Feld, “Fluorescence tomography imaging in turbid media using early-arriving photons and Laplace transforms,” Proc. Natl. Acad. Sci. USA. **94**, 8783–8788 (1997). [CrossRef] [PubMed]

*β*could be mathematically interpreted as the time-independent coupled diffusion equations with absorption coefficients of (

*μ*(

_{aν}**r**)

*c*+

*β*≥0(

*ν*∈{x,m}). In addition, we also know that

*S*(

_{m}**r**,

**r**

*,*

_{s}*β*)=

*c*Φ

*(*

_{x}**r**,

**r**

*,*

_{s}*β*)

*ημ*(

_{af}**r**)/[1+

*βτ*(

**r**)], so there should be [1+

*βτ*(

**r**)]≥0. Considered the two conditions together, the choice of the transform-factor

*β*should meet the condition that

*β*≥max{-

*μ*, -1/

_{aν}c*τ*}, so that the time-independent diffusion model is physically meaningful as well mathematically stable. To investigate the effect of transform-factors on reconstructed images, a variety of transform-factors are empirically employed.

*i.e.*the fluorescent yield and lifetime of the probe are to be recovered. The reconstructed quantity

*x*(

**r**,

*β*) is a function of the fluorescence yield, lifetime and transform-factor

*β*, so employing a pair of factors

*β*

_{1},

*β*

_{2}and the corresponding reconstructed

*x*(

**r**,

*β*

_{1}) and

*x*(

**r**,

*β*

_{2}), we can get the profiles of yield and lifetime simultaneously.

## 3. Validations

*Z*=6mm,

*Z*=16mm,

*Z*=26mm and

*Z*=36mm). On every layer 16 source-detector pairs are distributed at equal spacing. When 64 sources illuminate the surface in serial, 64 detectors will collect the photons in parallel. This leads to a total of 64×64 time-resolved measurements. A finite element method with 52377 nodes and 97200 elements (prisms) is used to solve the diffusion equations. The process is implemented in MATLAB.

_{ax,m}=0.035mm

^{-1}and μ′

_{sx,m}=1.0mm

^{-1}for the excitation and emission wavelengths, and the fluorescent properties of background are ημ

_{af}=0.001 mm

^{-1}and τ=100 ps. These values are in the range of the optical properties for in vivo muscle [28

28. F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. Depeursinge, “IN vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. **38**, 4939–4950(1999). [CrossRef]

### Case1: different yield and lifetime

*Z*=22mm) and (X=5mm, Y=0mm,

*Z*=22mm), respectively, as shown in Fig. 1(a). Different fluorescence properties of (ημ

_{af1}=0.003mm

^{-1},

*τ*

_{1}=200 ps) and (ημ

_{af2}=0.002mm

^{-1},

*τ*

_{2}=300 ps) are assumed for the two targets to investigate the ability of the algorithm to discern the difference of the fluorescent properties within the targets. Fig. 2 shows the original and reconstructed 2-D slice images for the fluorescent yield and lifetime at

*Z*=22mm. The reconstructed images show a reasonable fidelity in the position and size. More precisely, the reconstructed target sizes in the yield images are approximately the same as the original one while reconstructed target sizes in lifetime images are slightly larger than the true ones. However, the grayscale reconstruction is under-estimated for both the parameters.

### Case2: quantitativeness

^{1,2}

_{af}=0.0015mm

^{-1}, τ

^{1,2}=150 ps), 2:1(ημ

^{1,2}

_{af}=0.002mm

^{-1}, τ

^{1,2}=200 ps), 3:1(ημ

^{1,2}

_{af}=0.003mm

^{-1}, τ

^{1,2}=300 ps), 4:1(ημ

^{1,2}

_{af}=0.004mm

^{-1}, τ

^{1,2}=400 ps) and 5:1(ημ

^{1,2}

_{af}=0.005mm

^{-1}, τ

^{1,2}=500 ps). A measure, referred to as quantitativeness ratio, is introduced for the evaluation, which defined as the ratio of the peak value of the reconstructed yield (lifetime) in the X-profile to original value. Fig. 3(a) presents the original and reconstructed cross-sectional images of 3:1 contrast for target 1 and target 2 at

*Z*=22mm, Fig. 3(b) demonstrates their profiles along X-axis. The quantitativeness ratio of reconstructed yield and lifetime as a function of the contrast between the target and the background are given in Fig. 3(c). It is shown that the quantitativeness ratio of the reconstructed yield consistently decreases with the increase of the target contrast, while a decrease at the lower contrast is observed for the lifetime reconstruction. For both the yield and lifetime reconstructions, the quantitativeness ratios are over 65% for all the cases. In addition, it is qualitatively seen from Fig. 3 (a) that the spatial resolution of yield is superior to that of lifetime.

### Case3: the spatial- resolution

_{af}=0.002mm

^{-1}and τ=300 ps. Fig.4 (a) illustrates the reconstructed images for the CCS of 10mm and 8mm. It can be seen from Fig.4 (b) that the valley value between two targets in the reconstructed X-profile

*υ*(

*x*) increases with the CCS deceasing from 12mm to 6mm, exhibiting an increasing degradation in the spatial-resolution. To quantify the spatial-resolution performance, we define

*R*=(

_{υ}*υ*

_{max}-

*υ*(0))/(

*υ*

_{max}-

*υ*

_{min}),

*υ*∈(

*ημ*,

_{af}*τ*), where

*υ*

_{max}=max[

*υ*(

*x*)],

*υ*

_{min}=min[

*υ*(

*x*)], as the measure. With the definition,

*R*=1 and

_{υ}*R*=0 represent the maximal and minimum spatial-resolution, respectively. For the cases here,

_{υ}*R*at the CCS of 12mm, 10mm, 8mm and 6mm are calculated as 0.894, 0.590, 0.298, 0.132 and 0.0757, 0.390, 0.123, 0.0734 for the yield and lifetime, respectively. The results show that even the edge-to-edge separation of the two targets decreases to zero,

_{υ}*i.e.*CCS=6mm, a feeble separation is still observable. Overall, the spatial-resolution of the reconstructed yield is better than that of the reconstructed lifetime.

### Case 4: different positions in Z-axis

*Z*=25mm, while the X and Y coordinates are the same as those of Target 1 in Fig.1 (a). The reconstructed cross-sectional images of the yield and lifetime at

*Z*=24mm,

*Z*=23mm and

*Z*=22mm, respectively, are shown in Fig.5 (a). The figure shows that with Z increasing, the radius of target 2 increases but that of target 3 decreases. The variations in the target sizes correctly reflect the real situation. The Fig.5 (b) shows the longitudinal sectional images of the yield and lifetime at Y=0mm. To test the ability of algorithm to reconstruct target’s locations, we restrict two regions of interest with a radius of 5mm and the same centers as Target 2 and 3, and then define

*υ*

_{(i)}∈(

*μη*

_{af(i)},

*τ*

_{(i)}) to calculate the reconstructed target centers for quantifying the positional change. Table 1 gives the original and reconstructed “centers of gravity” of target 2 and target 3. The results show a very faithful localization performance of the algorithm.

### Case 5 Noise-robustness

*i.e.*, the Laplace transforms of the time-resolved data, as an additive Gaussian random variable with a standard deviation proportional to the data-type:

*σ*(

_{υ}**r**

*,*

^{i}_{d}**r**

*,*

^{j}_{ss}*β*)=

*I*(

_{υ}**r**

*,*

^{i}_{d}**r**

*,*

^{j}_{s}*β*)(1+10

^{-χ/20}

*R*), where

_{N}*χ*is the SNR and

*R*is the standard normally distributed random number. In our study we empirically selected a pair of factors

_{N}*β*=∓1/[1/(

_{L}*μ*

^{(B)}

*)+1/(*

_{ax}c*μ*

^{(B)}

*)+*

_{am}c*τ*

^{(B)}], where the superscript

*B*denotes background. A variety of transform-factors are employed, these values change with a step of ∓0.1

*β*. Here we just show the reconstructed images based on two pairs of transform-factors and two kinds of SNR for the limitation in paper length. Fig.6 (a) illustrates the reconstructed cross-sectional images with SNRs of 35dB and 25dB, and a pair of transform-factors ∓0.5

_{L}*β*. Fig.6 (b) shows the cross-sectional images with SNRs of 35dB and 25dB and a pair of transform-factors ∓

_{L}*β*. The results reveal that the image quality with ∓

_{L}*β*is improved evidently and the reconstruction of the fluorescent yield is more robust to the noise. More detailed investigation also shows that the quality of reconstructed images improves with increase in the difference between the two transform-factors. It seems that

_{L}*β*=∓1/[1/(

_{L}*μ*

^{(B)}

*)+1/(*

_{ax}c*μ*

^{(B)}

*)+*

_{am}c*τ*

^{(B)}] are extremum, since reconstructions with a transform-factor larger than 1/[1/(

*μ*

^{(B)}

*)+1/(*

_{ax}c*μ*

^{(B)}

*)+*

_{am}c*τ*

^{(B)}] or smaller than -1/[1/(

*μ*

^{(B)}

*)+1/(*

_{ax}c*μ*

^{(B)}

*)+*

_{am}c*τ*

^{(B)}] always fails to produce the correct images.

## 4. Discussion and conclusions

29. J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Full adaptive finite element based tomography using tetrahedral dual-meshing for fluorescence enhanced optical imaging in tissue,” Opt. Express **15**, 6955–6975 (2007). [CrossRef] [PubMed]

31. S. C. Davis, H. Dehghani, J. Wang, S. D. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express **15**, 4066–4082(2007) [CrossRef] [PubMed]

*β*=∓1/[1/(

*μ*

^{(B)}

*)+1/(*

_{ax}c*μ*

^{(B)}

*)+*

_{am}c*τ*

^{(B)}], has shown an optimal noise-robustness.

## References

1. | R. Weissleder, C.H. Tung, U. Mahmood, and A. Bogdanov, “In vivo imaging with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. |

2. | V. Ntziachristos, C-H Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. |

3. | Achilefu, R. Dorshow, J. Bugaj, and R. Rajapopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. |

4. | T.F. Massoud and S.S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. |

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6. | S.R. Cherry, “In vivo molecular and genomic imaging: new challenges for imaging physics,” Phys. Med. Biol. |

7. | H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulation,” Appl. Opt. |

8. | A.B. Milstein, S. Oh, K.J. Webb, C.A. Bouman, Q Zhang, D.A. Boas, and R.P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. |

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

10. | M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: Near-infrared fluorescence tomography,” Pans. |

11. | A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys. |

12. | B. Yuan and Q. Zhu, “Separately reconstructing the structural and functional parameters of a fluorescent inclusion embedded in a turbid medium,” Opt. Express |

13. | A. B Milstein, J.J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman, and K. J. Webb, “Fluorescence optical diffusion tomography using multiple-frequency data”, Opt. Soc. Am. A |

14. | A. Soubret and Vasillis Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. |

15. | A.T.N. Kumar, J. Skoch, B.J. Bacskai, D.A. Boas, and A.K. Dunn, “Fluorescent-lifetime-based tomography for turbid media,” Opt. Lett. |

16. | F. Gao, H.J. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomogrphy,” Opt. Express |

17. | V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, A. Sardini, P. M. W. French, and S. R. Arridge “Fluorescence lifetime tomography by using time-gated data acquisition,” Appl. Opt. |

18. | S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express |

19. | F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. |

20. | H.J. Zhao. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, “Time-resolved optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. |

21. | E.M.C. Hillman, J.C. Hebden, M. Schweiger, H. Dehghani, F.E.W. Schmidt, D.T. Delpy, and S.R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. |

22. | F. Gao, Y. Tanikawa, H.J. Zhao, and Y. Yamada, “Semi-three-dimensional algorithm for time-resolved diffuse optical tomography by use of the generalized pulse spectrum technique,” Appl. Opt. |

23. | V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. |

24. | J. Wu, “Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging of biological tissues,” J. Opt. Soc. Am. A |

25. | A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging |

26. | A.C Kak and M. Slaney, |

27. | J Wu, L. Perelman, R. R. Dasari, and M. S. Feld, “Fluorescence tomography imaging in turbid media using early-arriving photons and Laplace transforms,” Proc. Natl. Acad. Sci. USA. |

28. | F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. Depeursinge, “IN vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. |

29. | J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Full adaptive finite element based tomography using tetrahedral dual-meshing for fluorescence enhanced optical imaging in tissue,” Opt. Express |

30. | D. Wang, X. Song, and J. Bai, “Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution,” Opt. Express |

31. | S. C. Davis, H. Dehghani, J. Wang, S. D. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

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

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

(170.6920) Medical optics and biotechnology : Time-resolved imaging

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: March 13, 2008

Manuscript Accepted: April 29, 2008

Published: May 5, 2008

**Virtual Issues**

Vol. 3, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Limin Zhang, Feng Gao, Huiyuan He, and Huijuan Zhao, "Three-dimensional scheme for time-domain fluorescence molecular tomography based on Laplace transforms with noise-robust factors," Opt. Express **16**, 7214-7223 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-10-7214

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### References

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