## Time resolved fluorescence tomography of turbid media based on lifetime contrast

Optics Express, Vol. 14, Issue 25, pp. 12255-12270 (2006)

http://dx.doi.org/10.1364/OE.14.012255

Acrobat PDF (346 KB)

### Abstract

A general linear model for time domain (TD) fluorescence tomography is presented that allows a lifetime-based analysis of the entire temporal fluorescence response from a turbid medium. Simulations are used to show that TD fluorescence tomography is optimally performed using two complementary approaches: A direct TD analysis of a few time points near the peak of the temporal response, which provides superior resolution; and an asymptotic multi-exponential analysis based tomography of the decay portion of the temporal response, which provides accurate localization of yield distributions for various lifetime components present in the imaging medium. These results indicate the potential of TD technology for biomedical imaging with lifetime sensitive targeted probes, and provide useful guidelines for an optimal approach to fluorescence tomography with TD data.

© 2006 Optical Society of America

## 1. Introduction

1.
Hintersteiner*et al*, “In-vivo detection of amyloid-beta deposits by near-infrared imaging using an oxazinederivative probe,” Nat. Biotechnol. **23**, 577 (2005). [CrossRef] [PubMed]

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

4. M. A. Oleary, D. A. Boas, X.D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**, 158 (1996). [CrossRef]

5. A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys. **48**, 1701–1720 (2003). [CrossRef]

7. K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, “Optical computed tomography in a turbid medium using early arriving photons,” J. Biomed. Opt. **5**144 (2000). [CrossRef] [PubMed]

4. M. A. Oleary, D. A. Boas, X.D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**, 158 (1996). [CrossRef]

5. A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys. **48**, 1701–1720 (2003). [CrossRef]

6. B. B. Das, F. Liu, and R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. **60**, 227 (1997). [CrossRef]

11. G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, “System irf impact on fluorescence lifetime fitting in turbid medium,” Proc. SPIE **5699**, 263 (2005). [CrossRef]

11. G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, “System irf impact on fluorescence lifetime fitting in turbid medium,” Proc. SPIE **5699**, 263 (2005). [CrossRef]

17. P. R. Sevin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Biol. **7**, 730–734, (2000). [CrossRef]

18. P. I. H. Bastiaens and A. Squire, “Fluorescence lifetime imaging microscopy: spatial resolution biochemical processes in the cell,” Trends Cell. Biol. **9**, 48–52 (1999). [CrossRef] [PubMed]

19. B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. **8**, 368–375 (2003). [CrossRef] [PubMed]

13. A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence lifetime-based tomography for turbid media,” Opt. Lett. **30**, 3347–3349 (2005). [CrossRef]

*inverse*Laplace transform (which is equivalent to multi-exponential fitting for a few discrete lifetimes) to reconstruct the decay portion of the DFTR. However, the restriction of this method to the decay portion excludes the information from the earlier portion of the DFTR, which is characterized by a better signal-to-noise (SNR) ratio, and may also contain useful spatial information. It is thus imperative to seek an approach that incorporates the rising and peak portions of the DFTR data into the analysis. In this work, we develop a theoretical formalism that allows a lifetime-based separation of fluorescence yield distributions using the entire TD data. In order to evaluate the optimal choice of temporal measurements for tomography using the direct TD approach, we use a singular value decomposition analysis [29

29. E. E. Graves, J. P. Culver, J. Ripoll, R. Wessleder, 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]

30. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. **26**, 701–703 (2001). [CrossRef]

27. A. 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,” J. Opt. Soc. Am A **21**, 1035–1049 (2004). [CrossRef]

28. A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas, and B. J. Bacskai, “Time resolved fluorescence imaging in diffuse media,” Proc. of SPIE **6009**60090Y, (2005). [CrossRef]

## 2. Theoretical development

*η*(

**r**) and τ(

**r**)=1/Γ(

**r**), respectively. For tomography, optical sources and detectors are arranged on the surface of the imaging medium. The fluorescence intensity measured at a detector point

**r**

_{d}at time

*t*for an impulsive excitation at source position

**r**

_{s}and

*t*=0 can be written in the standard way as a double convolution over time, of the source and emission Green’s functions (omitting experimental scaling factors for simplicity):

*G*

^{x}and

*G*

^{m}denoting the source and emission Green’s functions, which depend on the net absorption and scattering coefficients (background+fluorophore) at the excitation and emission wavelengths,

**r**) and

**r**). The above expression assumes a single absorption and re-emission event due to the fluorophore. But this does not prevent the inclusion of multiple absorption of the excitation and emission light by fluorophores in the background medium, which can be incorporated by obtaining the

*G*

^{(x,m)}as solutions to the diffusion or transport equations at the excitation and emission wavelengths with the net absorption including the fluorophore absorption at these wavelengths.

*W*

^{B}can be pre-calculated with a knowledge of background optical properties, the advantage of Eq. (4) over Eq. (2) is that only a single time integral is left for the tomographic recovery of the yield and lifetime distributions [20]. A more useful form of this expression is realized if the fluorophores within the medium are described as multiple distributions,

*η*

_{n}(

**r**), corresponding to discrete lifetime components, τ

_{n}=1/Γ

_{n}. We then get, for the weight function of each lifetime component,

_{n}>τ

_{a}(=1/νμ

_{a}(

**r**)), (see Section 2.1) Eq. 5 can be expressed in a more elegant way that also reveals the connection with previously developed asymptotic lifetime-based-tomography [13

13. A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence lifetime-based tomography for turbid media,” Opt. Lett. **30**, 3347–3349 (2005). [CrossRef]

*G*

^{(x,m)}, which is a rigorous description of light transport in a turbid medium [21

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

**s**,

**s**′) is the scattering phase function. The source terms are dropped from the excitation RTE on the basis that the fluence is calculated away from the source location, and similarly from the emission RTE, given that only a single fluorophore emission event is considered in accordance with the Born approximation initially made in Eq. 2. (Multiple absorption of the excitation and emission light by the fluorophore is still incorporated in the total absorption at these wavelengths viz.,

*gradient*of the absorption coefficient, ∇

**r**), and independent of

**r**) itself. Thus,

**r**)-Γ

_{n}/

*ν*, which is positive under the long lifetime condition viz., Γ

_{n}<

**r**), it is then easily verified using Eq. (8) that

18. P. I. H. Bastiaens and A. Squire, “Fluorescence lifetime imaging microscopy: spatial resolution biochemical processes in the cell,” Trends Cell. Biol. **9**, 48–52 (1999). [CrossRef] [PubMed]

*A*

_{n}depend on time, in addition to the source and detector locations. The amplitudes define a linear inversion problem for the yield distributions of each lifetime component:

*A*

_{n}(

*t*) has a non-trivial time evolution that is determined by the size and optical properties of the imaging medium. In Figure 1, the temporal evolution of

*A*(

*t*) is shown for infinite slabs of thicknesses 2cm and 10cm, with a single 2mm

^{3}fluorophore inclusion of 1ns lifetime embedded at the center of the slab. Furthermore, the net fluorescence signal calculated using Eqs. (3) and (11–12) is compared with the fluorescence signal computed directly using Eqs. (1–2), to confirm the accuracy of the effective absorption model in Eq. (10). The above equations are also applicable to phosphorescence signals from diffuse media [16

16. S. V. Apreleva, D. F. Wilson, and S. A. Vinogradov, “Feasibility of diffuse optical imaging with long-lived luminescent probes,” Opt. Lett. **31**, 1082–1084 (2006). [CrossRef] [PubMed]

*A*(

*t*) can be approximated as a step function in time.

**Asymptotic limit**: From Eq. (10), it is clear that the weight function for each lifetime component is an average over a time

*t*of the background sensitivity function

_{D}denote the timescale for the evolution of

*t*>τ

_{D}, the average over

*W*will then become time independent and reduce to a CW sensitivity function, which we denote by

^{B}_{n}*W̅*

_{n}. We are thus lead to asymptotic lifetime-based tomography, which was derived previously using complex integration (see Eqs. (3) and (4) in Reference 13

13. A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence lifetime-based tomography for turbid media,” Opt. Lett. **30**, 3347–3349 (2005). [CrossRef]

*j*to denote measurement index that labels each sourc-edetector (S-D) pair. For times longer than τ

_{D}, the decay amplitude becomes time-independent, so that the amplitude for each lifetime component can be recovered asymptotically using multi-exponential fits. These amplitudes constitute a derived data set (inverse Laplace transform) for the inversion of the yield distributions:

### 2.1. Conditions for asymptotic recovery of intrinsic fluorophore lifetimes

_{a}=(

*νµ*

_{a})

^{-1}, which is the asymptotic decay time of the diffuse temporal response (DTR) at the excitation wavelength, in the limit of homogenous semi-infinite media [24

24. M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

_{D}, of the DTR from a finite sized imaging medium, which includes the effect of boundaries. It is known that the presence of boundaries reduces the decay time [24

24. M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

25. J. C. Haselgrove, J. C. Schotland, and J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. **31**, 2678–2683 (1992). [CrossRef] [PubMed]

_{D}<τ

_{a}. Since the DFTR is a convolution of the fluorescence decay with the DTR, two scenarios emerge for a lifetime based analysis of TD fluorescence data from diffuse media:

**Strong condition**: τ

_{n}>τ

_{a}. Since τ

_{a}>τ

_{D}, this guarantees that lifetimes can be measured asymptotically, irrespective of tissue optical properties and medium boundaries. Furthermore, the multi-exponential model presented in Eqs. (11–12) is valid.

**Weak condition**: τ

_{a}>τ

_{n}>τ

_{D}. Lifetimes can still be measured asymptotically, but the reduced absorption model in Eq. (10) is no longer valid. The more general expression for the weight function, viz., Eq. (5), should instead be used for both the direct TD and asymptotic reconstructions.

*µ*

_{a}>0.1

*cm*

^{-1}corresponds to τ

_{a}<0.5

*ns*). In applications with small volumes as in small animal imaging [12

12. S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, and S Achilefu, “Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice,” J. Biomed. Opt. **10**, 054003-1 (2005). [CrossRef]

_{D}for a range of tissue optical properties can be found in Reference 13

**30**, 3347–3349 (2005). [CrossRef]

25. J. C. Haselgrove, J. C. Schotland, and J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. **31**, 2678–2683 (1992). [CrossRef] [PubMed]

_{a}, in evaluating the above conditions. The above two simple rules dictate the condition for measuring intrinsic lifetimes from surface fluorescence decays for arbitrary diffuse imaging media. Note that the

*average*decay time on the surface might itself change due to factors that affect the amplitude of individual lifetime components (e.g., thickness of autofluorescence layers [31

31. K. Viswanath and M. Mycek, “Do fluorescence decays remitted from tissues accurately reflect intrinsic fluorophore lifetimes?,” Opt. Lett. **29**, 1512–1514 (2004). [CrossRef]

## 3. Singular value analysis of the time domain weight function

*mm*

^{3}(1mm×1mm×2mm). The temporal points were chosen to be 200

*ps*apart, corresponding to the typical minimum gate width in time-gated detection techniques [10

10. G. M. Turner, G Zacharakis, A. Sourbet, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. **30**, 409 (2005). [CrossRef] [PubMed]

**30**, 3347–3349 (2005). [CrossRef]

*ns*. To begin with, consider performing tomography using Eq. 14 with all S-D pairs and a single time point. What is the location for this time point for an optimal reconstruction? To answer this question, the singular value spectra for

*W*

_{n}evaluated at various time points were calculated. The five spectra with the highest values are plotted in Fig 3(a), and the number of singular values,

*N*

_{σ}, above a chosen noise threshold of 10

^{–14}is plotted in the inset of Fig. 3(b). It is seen that the spectrum for the time gate near the peak has the highest number of singular values above the noise threshold. It is noteworthy that the slope of the singular value spectrum is lowest for the earliest time, and increases for later times. This could be attributed to the narrower spatial sensitivity profile sampled by the early arriving photons. However, the higher signal level (and the best SNR, in the presence of shot noise) near the peak of the DFTR overcomes the faster decay of the spectrum, resulting in a larger

*N*

_{σ}near the peak. We thus conclude that tomography with a single time gate is optimally performed with a time point near the peak of the DFTR. Note the linearity of

*N*

_{σ}in the exponential decay region (red curve in the inset of Fig 3(b)). This could be attributed to the fact that the SVD spectra are also approximately exponential, as evident from the log plot in Fig. 3(a), so that the intercept of diag(S) at fixed noise threshold depends linearly on time.

*N*

_{σ}, was determined. It turns out that one of the time points was again near the peak of the DFTR and the other was located near the rise portion of the DFTR. In the same way, the location and

*N*

_{σ}for multiple combinations of time points were determined. In Figure 3(b),

*N*

_{σ}is plotted as a function of the number of time points used. It seen that the proportional increase in

*N*

_{σ}diminishes rapidly after the first 3 or 4 temporal measurements. Also shown in the inset in Fig 3(b) are the first five optimal time points on a representative DFTR on the surface, which are located near the initial portion of the DFTR before the beginning of the fluorescence decay. It was determined that additional time points were located near the decay portion of the DFTR and added little to

*N*

_{σ}.

*not*ideal for tomography with the long time decay data. (When the lifetimes are widely separated, the shorter lived components may be suppressed by reconstructing later delays [16

16. S. V. Apreleva, D. F. Wilson, and S. A. Vinogradov, “Feasibility of diffuse optical imaging with long-lived luminescent probes,” Opt. Lett. **31**, 1082–1084 (2006). [CrossRef] [PubMed]

## 4. Tomography using direct TD and asymptotic approaches

*intrinsic*fluorescence lifetime(s) and the corresponding decay amplitude(s) from the asymptotic tail. (2) Reconstruct the individual yield distributions for each decay component using the decay amplitudes for all S-D pairs. (3) Reconstruct the yield distributions for each lifetime component using a few time points near the rise and the peak of the DFTR. These three steps reduce the computational complexity involved in a brute-force reconstruction of a large temporal data set, while retaining the most complete information possible from a TD experiment.

*U*,

*S*,

*V*of the weight matrix

*W*

_{n}. Denoting the measurement vector by

*Y*and the image by

*X*, the inversion takes the following typical form for under-determined problems

*Y*=

*W*

_{n}

*X*:

*α*=max{diag(

*W*

_{n})} and the regularization parameter λ is typically between 10

^{-5}and 10

^{-3}. Three different reconstructions were performed, namely, CW, direct TD, and asymptotic. The CW reconstructions were performed using the time integrated TD data. The direct TD reconstructions used Eq. (14) with a set of 4 time points on the rising edge of the DFTR, following the SVD analysis results of Fig. 3. The asymptotic TD reconstruction was performed using the amplitudes obtained from a multi-exponential analysis of the decay portion in Eq. 15.

*η*

_{1}and

*η*

_{2}for the 1ns and 1.5ns lifetimes, the cross-talk X was estimated to quantify the separability of the two inclusions based on lifetimes. If Ω

_{1}denotes a chosen region-of-interest around the known location of the 1ns inclusion, then X

^{1ns}=max[

*η*

_{2}(Ω

_{1})]/max[

*η*

_{1}(Ω

_{1})]. The yield cross talk for the 1.5ns component was similarly evaluated. The CNR vs FWHM plot is shown in Fig. 4 for the 1ns lifetime inclusion, for the case with 7mm lateral separation between the inclusions. It is clear that the TD reconstruction shows a dramatic improvement in the CNR and FWHMover the asymptotic reconstruction, and an improvement over the resolution of the CWcase. The CNR improvement is evidently due to the better SNR of the peak portion of the DFTR compared to the asymptotic tail. The FWHM improvement of the TD over CWis due to the tomographic separation of the yield distributions for the lifetime components. Thus, for fixed CNR, the lifetime based TD reconstruction will have superior resolution compared to the asymptotic and CW reconstructions. However, the cross-talk, (which is the reconstructed amplitude of the 1.5ns inclusion at the location of the 1ns inclusion) is significantly higher for TD than the asymptotic case, and is attributable to the non-diagonal nature of the TD portion of the forward problem in Eq. (14). We note that the crosstalk for the asymptotic approach will depend on the separability of the lifetimes from the multi-exponential fits of raw experimental data, an aspect that will be explored in future work.

*log*

_{10}(CNR) was near 1. This is necessary to properly account for the difference in the noise characteristics of the different methods. (For example, CW has the best SNR, and should thus be the least regularized.) To visualize crosstalk easily, the yield images for the 1ns and 1.5ns components for the TD and asymptotic approaches were assigned red and green colors in an RGB colormap of a single image. The degree of crosstalk is thus revealed as a mixture of these two colors (e.g., yellow implies 100% crosstalk). Thus, CW reconstructions have no lifetime information so that they are shaded in yellow. It is clear from Fig. 5 that the TD reconstruction has superior resolution but significantly more cross-talk than the asymptotic reconstructions, as can also be seen in the intensity plots in the bottom panel. For small target separations, the cross talk of the TD method proves detrimental to its accuracy, whereas the asymptotic case recovers the localizations accurately even for 3mm separation. Thus it can be said that the direct TD approach using optimal time points provides more precise (better-resolved) reconstructions, and is useful when the targets are well separated, whereas the asymptotic reconstructions are more accurate but less precise (less-resolved). In Figure 6, the reconstructions are shown with the targets located axially, i.e., along the S-D axis. The advantage of the lifetime based asymptotic reconstruction is even more evident in this case, as the localizations of the two lifetimes are not reproduced either for the CW or the direct TD reconstructions.

_{D}, which is near 0.4ns for the present simulation. (corresponding to

*µ*

_{a}=0.1). As the mean lifetime becomes much larger than τ

_{D}, the cross talk will also increase, diminishing the separability of the corresponding yield distributions. This is due to the fact that the elements of the first row of the TD weight matrix in Eq. (14) are almost identical for the early time points, when τ

_{n}≫τ

_{D}. To study this quantitatively, the simulations in Fig. 7 (b) and (c) were repeated for a range of mean lifetimes, with fixed lifetime separation of 0.5ns and the crosstalk was estimated for each case. In Fig. 7(e), the cross talk of the direct TD approach is plotted as a function of the mean lifetime of the inclusions, indicating the large range of lifetimes for which direct TD reconstructions can benefit from lifetime contrast.

## 5. Conclusions

## Acknowledgments

## References and links

1. |
Hintersteiner |

2. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light,” Nat. Biotechnol. |

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

4. | M. A. Oleary, D. A. Boas, X.D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. |

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

6. | B. B. Das, F. Liu, and R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. |

7. | K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, “Optical computed tomography in a turbid medium using early arriving photons,” J. Biomed. Opt. |

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

9. | D. Hall, G. Ma, F. Lesage, and Y. Wang, “Simple time-domain optical method for estimating the depth and concentration of a fluorescent inclusion in a turbid medium,” Opt. Lett. |

10. | G. M. Turner, G Zacharakis, A. Sourbet, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. |

11. | G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, “System irf impact on fluorescence lifetime fitting in turbid medium,” Proc. SPIE |

12. | S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, and S Achilefu, “Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice,” J. Biomed. Opt. |

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

14. | D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, and J. Opt. Soc. Am. A, “Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue,” J. Opt. Soc. Am. |

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

16. | S. V. Apreleva, D. F. Wilson, and S. A. Vinogradov, “Feasibility of diffuse optical imaging with long-lived luminescent probes,” Opt. Lett. |

17. | P. R. Sevin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Biol. |

18. | P. I. H. Bastiaens and A. Squire, “Fluorescence lifetime imaging microscopy: spatial resolution biochemical processes in the cell,” Trends Cell. Biol. |

19. | B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. |

20. |
Equation (4) is a generalized form of a semi-analytical expression for an infinite medium given in Reference 9, as can be checked by setting |

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

22. | S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960). |

23. | F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in timedomain fluorescence molecular tomography,” Opt. Express |

24. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

25. | J. C. Haselgrove, J. C. Schotland, and J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. |

26. | A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, “On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales,” J. Phys. Chem. B. |

27. | A. 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,” J. Opt. Soc. Am A |

28. | A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas, and B. J. Bacskai, “Time resolved fluorescence imaging in diffuse media,” Proc. of SPIE |

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

30. | J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. |

31. | K. Viswanath and M. Mycek, “Do fluorescence decays remitted from tissues accurately reflect intrinsic fluorophore lifetimes?,” Opt. Lett. |

**OCIS Codes**

(170.3650) Medical optics and biotechnology : Lifetime-based sensing

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

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

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: October 6, 2006

Revised Manuscript: November 27, 2006

Manuscript Accepted: November 29, 2006

Published: December 11, 2006

**Virtual Issues**

Vol. 2, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Anand T. Kumar, Scott B. Raymond, Gregory Boverman, David A. Boas, and Brian J. Bacskai, "Time resolved fluorescence tomography of turbid media based on lifetime contrast," Opt. Express **14**, 12255-12270 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12255

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### References

- Hintersteiner et al., "In-vivo detection of amyloid-beta deposits by near-infrared imaging using an oxazinederivative probe," Nat. Biotechnol. 23, 577 (2005). [CrossRef] [PubMed]
- V. Ntziachristos, J. Ripoll, L. V. Wang, R. Weissleder, "Looking and listening to light," Nat. Biotechnol. 23, 314 (2005).
- E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeter resolution for small animal imaging," Med. Phys. 30, 901 (2003). [CrossRef] [PubMed]
- M. A. Oleary, D. A. Boas, X.D. Li, B. Chance, and A. G. Yodh, "Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158 (1996). [CrossRef]
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