## A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomography

Optics Express, Vol. 14, Issue 16, pp. 7109-7124 (2006)

http://dx.doi.org/10.1364/OE.14.007109

Acrobat PDF (507 KB)

### Abstract

Fluorescence diffuse optical tomography (DOT) has attracted many attentions from the community of biomedical imaging, since it provides effective enhancement in imaging contrast. This modality is now rapidly evolving as a potential means of monitoring molecular events in small living organisms with help of molecule-specific contrast agents, referred to as fluorescence molecular tomography (FMT). FMT could greatly promote pathogenesis research, drug development, and therapeutic intervention. Although FMT in steady-state and frequency-domain modes have been heavily investigated, the extension to time-domain scheme is imminent for its several unique advantages over the others. By extending the previously developed generalized pulse spectrum technique for time-domain DOT, we propose a linear, featured-data image reconstruction algorithm for time-domain FMT that can simultaneously reconstruct both fluorescent yield and lifetime images of multiple fluorephores, and validate the methodology with simulated data.

© 2006 Optical Society of America

## 1. Introduction

01. D.Y. Patthankar, 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]

06. K. Licha, “Contrast agents for optical imaging,” Topics in Current Chemistry **222**, 1–29 (2002). [CrossRef]

10. V. Ntziachristos, C-H Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. **8**, 757–60 (2002). [CrossRef] [PubMed]

*in-vivo*; 2) optimizing drug and gene therapy and evaluating their effects at a molecular and/or cellular level; 3) assessing disease progression at a molecular pathological level, and 4) rapidly, reproducibly and quantitatively observing time-dependent experimental, developmental, environmental, and therapeutic influences on gene production in the same subject [15

15. S.R. Cherry, “In vivo molecular and genomic imaging: new challenges for imaging physics,” Phys. Med. Biol. **49**, R13–48 (2004). [CrossRef] [PubMed]

16. T.F. Massoud and S.S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. **17**, 545–580 (2003). [CrossRef] [PubMed]

01. D.Y. Patthankar, 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]

12. S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express **13**, 2263–2275 (2005). [CrossRef] [PubMed]

12. S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express **13**, 2263–2275 (2005). [CrossRef] [PubMed]

13. 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. **30**, 3347–3349 (2005). [CrossRef]

^{2+}] and pH

*etc*.) of fluorophores, while the multi-component analysis, where multiply sorts of molecular targets are traced by using different imaging reporter probes (components), potentially enables assessment of the multi-gene controlling mechanism in a disease progression as well as simultaneous observation of the multiple molecular/cellular events in a specific biochemical pathway [15

15. S.R. Cherry, “In vivo molecular and genomic imaging: new challenges for imaging physics,” Phys. Med. Biol. **49**, R13–48 (2004). [CrossRef] [PubMed]

16. T.F. Massoud and S.S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. **17**, 545–580 (2003). [CrossRef] [PubMed]

18. S.R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–93 (1999). [CrossRef]

18. S.R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–93 (1999). [CrossRef]

21. M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. **44**, 1699–1717 (1999). [CrossRef] [PubMed]

22. F. Gao, P. Poulet, and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from a three-dimensional model of time-resolved optical tomography,” App. Opt **39**, 5898–5910 (2001). [CrossRef]

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. **41**, 778–791 (2002). [CrossRef] [PubMed]

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. **41**, 778–791 (2002). [CrossRef] [PubMed]

21. M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. **44**, 1699–1717 (1999). [CrossRef] [PubMed]

22. F. Gao, P. Poulet, and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from a three-dimensional model of time-resolved optical tomography,” App. Opt **39**, 5898–5910 (2001). [CrossRef]

18. S.R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–93 (1999). [CrossRef]

21. M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. **44**, 1699–1717 (1999). [CrossRef] [PubMed]

22. F. Gao, P. Poulet, and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from a three-dimensional model of time-resolved optical tomography,” App. Opt **39**, 5898–5910 (2001). [CrossRef]

25. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by time-domain detection: methodology and phantom validation,” Phys. Med. Biol. **49**, 1055–1078 (2004). [CrossRef] [PubMed]

*in vitro*and

*in vivo*experiments [23–26

23. 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. **46**, 1117–1130 (2002). [CrossRef]

25. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by time-domain detection: methodology and phantom validation,” Phys. Med. Biol. **49**, 1055–1078 (2004). [CrossRef] [PubMed]

27. 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. **41**, 7346–7358 (2002). [CrossRef] [PubMed]

## 2. Methodology

### 2.1 Forward Model

*i.e.*the

*P*

_{1}approximation to the radiative transfer equation, has been prevalently employed as the photon-migration model, which is mathematically tractable either analytically or numerically [18

**15**, R41–93 (1999). [CrossRef]

*x*and

*m*denote the excitation and emission wavelengths, respectively; Φ

_{v}(

**r**,

**r**

_{s},

*p*) =

_{v}(

**r**,

**r**

_{s},

*t*)

*e*

^{-pt}

*dt*(

*v*∈[

*x,m*]) is the Laplace transform of the time-dependent photon density Φ

_{v}(

**r**,

**r**

_{s},

*t*) with a complex transform-factor

*p*; the optical parameters involved are the absorption coefficient μ

_{av}(

**r**), the reduced scattering coefficient μ′

_{sv}(

**r**) and the diffusion coefficient

*D*

_{v}(

**r**,

*t*) =

*c*/[3μ′

_{sv}(

**r**)]; the fluorescence parameters are the fluorescent yield ημ(

**r**) and lifetime τ(

**r**). These quantities, in general, are functions of the position vector

**r**. We employ uniquely the Robin boundary condition for the above equations

**K**= (1 +

*R*

_{f})/(1 -

*R*

_{f}) and

*R*

_{f}≈ -1.4399

*n*

^{-2}+ 0.7099

*n*

^{-1}+ 0.6681 + 0.0636

*n*is the internal reflection coefficient at the air-tissue boundary, with

*n*being the relative refractive index of tissue to air [28]. The measurable flux,

*i.e*., the data-type at the boundary site ξ

_{d}(

*d*= 1,2,…,

*D*)) and for the source site ζ

_{s}(

*s*= 1,2,…,

*S*), can be calculated by the Fick’s law with consideration of Eq.(2)

_{v}(

**r**,

**r**

_{s},

*p*) into finite elements: Φ

_{v}(

**r**,

**r**

_{s},

*p*) ≈

_{v}(

*n,p*)

*u*

_{n}(

**r**) =

**Φ**

_{v}(

*p*)

^{T}u(

**r**) with

**u**(

**r**) = [

*u*

_{1}(

**r**),

*u*

_{2}(

**r**), ⋯,

*u*

_{N}(

**r**)

^{T}and

*Φ*

_{v}(

*p*) = [Φ

_{v}(1,

*p*),Φ

_{v}(2,

*p*),⋯,Φ

_{v}(

*N,p*)]

^{T}being the shape functions and the Laplace-transformed photon density at the

*N*nodes of the FEM mesh, respectively, resulting in following matrix equation

**A**

_{v},

**B**and

**Q**

_{v}are given below

*i*and

*j*index the

*N*nodes of the mesh; ημ

_{af}(

*j*) and τ(

*j*) the fluorescent yield and lifetime at the

*j*-th meshing node, respectively.

### 2.2 Inverse model

_{m}(ξ

_{d}, ζ

_{s},

*p*) is the Laplace transform of the transient emission flux measured at boundary site ξ

_{d}and for excitation site ζ

_{s}, and

*G*

_{m}(ξ

_{d},

**r**,

*p*) the flux at ξ

_{d}for a source at

**r**, predicted by the diffusion equation at the emission wavelength

*x*(

**r**,

*p*)≈

*x*

_{n}(

*p*)

*u*

_{n}(

**r**) =

**x**

^{T}(

*p*)

**u**(

**r**) , where

**x**(

*p*) = [

*x*

_{1}(

*p*),

*x*

_{2}(

*p*),…,

*x*

_{N}(

*p*)]

^{T}, Eq.(6) can be discretized into a matrix equation

_{n}numerates all the elements that are joined at the

*n*-th node;

*Ḡ*

^{(Ωm)}

_{m}(ξ

_{d},

*p*) and

^{(Ωm)}

_{x}(ξ

_{d},

*p*) are the mean values of

*G*

_{m}(ξ

_{d},

**r**,

*p*) and Φ

_{x}(

**r**,ζ

_{s},

*p*) at the nodes of the element Ω

_{n}, respectively. By the sense of Eq. (8), the inverse procedure is based on the mesh nodes, which usually represents a significant reduction in the number of the unknowns, as compared with those that are based on the elements. This will be further explained in the next section.

**15**, R41–93 (1999). [CrossRef]

**39**, 5898–5910 (2001). [CrossRef]

*Γ*

^{(i)}(

*p*) is the

*i*-th element of

*Γ*(

*p*) and

**W**

^{(i)}(

*p*) the

*i*-th row of

**W**(

*p*), with (

*i*mod

*j*) denoting the modulus after

*i*divided by

*j*;

*M*is referred to as the ART-circulation number. The initial point

**x**

_{0}(

*p*) is set to what reflects the morphology of the background fluorescence as closely as possible. In principle, the ART sequentially projects a solution estimate onto the hyperplanes defined by the individual rows of the linear system. The relaxation parameter λ has a significant influence on the reconstruction and has been proven in a range of [0, 2] to make the algorithm converge to a point on the intersection of the governing equations that is nearest to the initial point [29]. The regularization strategy in the ART-based algorithms is accomplished by limiting the number of the iterations, whose choice is task-dependent and mandatory in presence of noise. The primary advantage of this implicit inversion method over the other schemes for solving underdetermined linear systems, such as the Levenberg-Marquardt, the truncated Singular Value Decomposition and the Conjugate-Gradient algorithms, is its near independence of memory occupation, since the rows of the weight matrix are successively employed during the solution process. However, the accuracy of the ART strongly relies on the initial guess of the unknowns. Sometimes, the prior information on the background as well as the targets is necessarily required to attain an image reconstruction of high-quality. To suppress the artifacts arising in the above ART inversion, a median filter operating on the adjacent nodes (

*i.e*., those that belong to the same element as the output node) is employed at the end of each circulation of the ART inversion.

#### 2.2.1 One-component case

*i.e.*the fluorescent yield ημ

_{af}(

**r**) and lifetime τ(

**r**) of the probe, is to be recovered. This can be explicitly done from the images of

*x*(

**r**,

*p*

_{1}) and

*x*(

**r**,

*p*

_{2}) by employing a pair of transform-factors:

*p*

_{1}and

*p*

_{2}, in the Laplace transforms

#### 2.2.2 Multi-component case

*N*

_{c}is the number of the components under investigation. The recovery of the 2

*N*

_{c}unknown distributions in Eq. (14) can in principle be achieved by using at least 2

*N*

_{c}transform-factors in the Laplace transforms:

*p*

_{1},

*p*

_{2},…,

*p*2

*N*

_{c}. For the two-component case, this leads to a set of four joint equations as follows

*x*(

**r**,

*p*

_{i}) (i = 1,2,…,2

*N*

_{c}) is obtained from solving Eq. (8) for each of the transform-factors. Let

**α**(

**r**), we can further obtain the four unknown distributions of the fluorescence parameters,

*i.e.*, ημ

_{af1}(

**r**), τ

_{1}(

**r**), ημ

_{af2}(

**r**) and τ

_{2}(

**r**), in terms of Eq. (16)

*N*

_{c}> 2, a similar procedure to the two-component case can be followed, which is mathematically more complex. It should be noted that within the above FEM framework Eq. (17) and Eq. (21) need to be solved successively for each node of the mesh, given the values of

**x**(

**r**) at the positions of the discrete nodes.

*N*

_{c}, real transform-factors are theoretically enough for the reconstruction of the 2

*N*

_{c}, independent unknowns in a

*N*

_{c}-component case, this may bring about considerable errors in the images due to the evident underestimation of

**x**(

**r**,

*p*) in Eq. (8). To overcome the adversity, more than 2

*N*

_{c}real transform-factors are in general employed, making Eq. (17) both overestimated and ill-posed. We therefore introduce an extended Levenberg-Marquardt method for the reliable solution, which is expressed as an optimization problem with the Tikhonov-Miller regularization as the follows [30

30. F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image and Vision Computing **16**, 703–712 (1998). [CrossRef]

**α**

_{B}(

**r**) is the value of

**α**(

**r**) calculated for the known background fluorescent properties at each node. The regularization matrix

**R**(

**r**)is chosen in such a way that

**R**

^{T}

**R**(

**RR**

^{T}) and

**M**

^{T}M (

**MM**

^{T}) commute each other, meaning that the two matrices have the same complete set of eigenvectors and the reversely-ordered eigenvalues,

*i.e.*,

*R*

_{M}is the rank of

**M**,

**u**

_{k}and

**v**

_{k}are the eigenvectors of

**MM**

^{T}and

**M**

^{T}

**M**belonging to the nonzero eigenvalue μ

_{k}. The solution to the above optimization problem can be readily found and leads to the matrix equation below

**r**implies that they needs to be solved for each FEM node, as indicated before. Mathematically, Eq. (22) or Eq. (23) seeks the filtered least-squares solution to Eq. (17) on the basis of its smoothness around the baseline (the known background properties). The smoothing weight is controlled by the regularizing factor β, whose value is task-dependent. Again, the median filtering is applied to

**α**(

**r**) images as well as the resultant ημ

*(*

^{af}**r**) and τ(

**r**) ones, respectively, for a further suppression of the artifacts.

## 3. Validations

*R*= 25 mm using simulated data, the principle of the methodology is applicable to three-dimensional models of the realistic geometry and the results represent an ideal experimental situation where the effects of systematic error on the reconstruction are ignored. To apply FEM, the domain is divided into 3750 triangles that join at 1951 nodes. 16 coaxial source-detector optodes (

*S*=

*D*= 16), are assumed at equal spacing around the annulus, of which the 16 detectors collect the exiting photons in parallel as the 16 sources illuminate the surface successively. This leads to a total of 256 time-resolved measurements. In our algorithm only measurements from the 9 detectors opposite to the illuminating source,

*i.e*. a set of 144 time-resolved data, are employed for the image reconstruction. It was found in our previous studies that such a configuration of data set can effectively reduce the difference in the order of the data magnitude and therefore significantly improve the image quality [19

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. **41**, 778–791 (2002). [CrossRef] [PubMed]

25. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by time-domain detection: methodology and phantom validation,” Phys. Med. Biol. **49**, 1055–1078 (2004). [CrossRef] [PubMed]

*e.g.*, from 3750 to 1951 in our case, as compared to a full element-based scheme. Figure 1 shows the FEM mesh of the domain and the deployment of the optodes used throughout the paper.

### 3.1 One-component case

*p*

^{1,2}= ∓0.1

*P*, where

*P*= 1/[1|μ

*c*+ 1/

*c*+ τ

^{(B)}], for the Laplace transforms of the time-domain forward and inverse models, and a circulation number

*M*= 20 as well as a relaxation parameter λ = 0.5 for the ART solution to Eq. (8), the derived linear system. The validation is firstly enforced by two numerical phantoms, for which the noiseless sets of the data-type

**Γ**(

*p*) are generated from the forward model with the same mesh as that to be used in the inverse model, to evaluate the intrinsic performance of the algorithm. In Phantom 1, four fluorescent circular targets with the same radius but different fluorescent properties are included to investigate the ability of the algorithm to discern the difference in the fluorescent properties of the targets, while Phantom 2 embeds two disks with same fluorescence parameters but different radiuses in order to probe the performance of the algorithm to reconstruct the target size. The optical properties of the targets in the two phantoms are the same as those of the background, which are set to μ

_{ax,m}= 0.035mm

^{-1}and μ′

_{sx,m}= 1.0mm

^{-1}for both the excitation and emission wavelengths. These values are in the range of the optical properties for

*in vivo*muscle [31

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

_{ax,m}=0.035 mm

^{-1}, μ′

_{sx,m}=1.0 mm

^{-1}, ημ

_{af}=0.005 mm

^{-1}and τ = 1000 ps , but a variable center-to-center separation (CCS), are placed along the X-axis, as shown in Fig. 4(a). Figure 4(b) presents the reconstructed images and Fig. 4(c) their profiles along the X-axis, for three different values of the CCS: 13 mm, 15mm and 17 mm. It is seen from the results that the two disks can still be resolved as the CCS is less than 13 mm (the valleys drops from the peaks by about 9% and 5% for the yield and lifetime, respectively), and the recovery of the fluorescent yield appears to have better resolution but poorer quantitativeness (only about 45% of the true value is reconstructed by the peaks) than the lifetime in terms of the profiles.

*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(ξ

_{d}, ζ

_{s},

*p*) = Γ

_{m}(ξ

_{d}, ζ

_{s},

*p*)10

^{-χ/20}, where

*d*= 1,2,…,

*D*,

*s*= 1,2,…,

*S*, and

*χ*is the SNR in decibels. Figure 5 illustrates the reconstructed images of the phantom a varying SNR:

*χ*= 35 dB , 40 dB and 45 dB . The results reveal that the noise-robustness of the algorithm is moderate and the reconstruction of the fluorescent yield is much more insensitive to the noise but less accurate in the quantitativeness than that of the lifetime.

### 3.2 Two-component case

*P*to +0.25

*P*with a step of 0.05

*P*is used for the Laplace transforms. This makes Eq. (17) overdetermined and ill-posed, with the condition number of its normal equation being 10

^{5}in the order of magnitude, for which the least-square solution can be approximately found with the Levenberg-Marquardt scheme. The reconstruction is performed with Gaussian noisy data of χ = 45 dB and a fixed regularizing factor β = 10

^{-3}. Figure 6 shows the original and reconstructed images for the two-component phantom, as well as their profiles along X-axis. It is observed that the reconstruction of the fluorescent yield correctly reflects the difference between the two components but both are underestimated, while the resultant images of the lifetime are somewhat distorted with the first component underestimated and the second overestimated.

## 4. Discussions and conclusions

*N*

_{c}targets, a proper choice of at least

*N*

_{c}transform-factor pairs would selectively pick up the signal weights that favor the different fluorescent emissions, and lead to a reliable separation among the fluorophores. A major disadvantages of the real-domain Lapalace transform is that it is now unclear if the information embedded in the time-resolved data can be extracted effectively, as with the Fourier transform, which mathematically complete along the imaginary domain.

*i.e.*,

_{d}, ζ

_{s},

*p*) and

_{d}, ζ

_{s},

*p*) are the data-types before and after probe injection, respectively;

_{d}, ζ

_{s},

*p*) the model prediction in terms of the background parameters. Differential imaging has been proved to be able to cancel many systematic and measuring noises and greatly improve the image quality [23–27

23. 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. **46**, 1117–1130 (2002). [CrossRef]

10. V. Ntziachristos, C-H Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. **8**, 757–60 (2002). [CrossRef] [PubMed]

26. Huijuan Zhao, Gao Feng, Yukari Tanikawa, Kazuhiro Homma, and Yukio Yamada, “Time-resolved optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. **43**, 1905–1916 (2005). [CrossRef]

*i.e.*, Eq. (23), has been promisingly applied for generating the reliable

**α**-images from the overdetermined but ill-conditioned linear system that are derived from the

**x**-images at a number of different tramsform-factors, with a constant regularizing-factor β. Furthermore, it has been shown that an L-curve criterion for determining β can better trade off between the data misfit ||

**x**(

**r**) -

**M**(

**r**)α(

**r**)|| and the solution-to-baseline seminorm ||

**R**(

**r**)[α(

**r**) - α

_{B}(

**r**)]||, but at a considerably increasing cost of computation. By comparison, we found that this regularized overestimation scheme can produce much higher quality of images than the simple matrix inversion using only four transform-factors. It is also pointed out here that, although we confine the fluorescent heterogeneities for the two components to the same regions for simplicity, it is unnecessary in the methodology for the fluorescence regions of the multiple probes to be the same as each other. The allowance for the maximal freedom of the probe distributions agrees with the fact that the multiple probes specifically bound to their respective biochemical molecules have little probability of the same fluorescent emission behavior, due to the intrinsic differences among the distributions of the molecule concentrations and among the efficacies of the fluorescent probes.

*p*,

*i.e.*, Eq. (1), can mathematically interpreted as the time-independent coupled diffusion equations with absorption coefficients of μ

_{av}+

*p*/

*c*(

*v*∈ {x, m}). Therefore, together with considering the exponentially decaying response of the fluorescent emission, the choice of the transform-factor

*p*should meet the condition that

*p*≥ max{- μ

_{av}

*c*, - 1/

*τ*} so that the time-independent diffusion model is physically meaningful as well mathematically stable. Besides this criterion, there is no theoretical guide to the choice of the transform-factor pairs. Therefore the transform-factor pairs used in the above examples are somewhat empirical. Nevertheless, the above numerical validations of the methodology with these empirically chosen transform-factor pairs have achieved our goal with considerable success. The selection of the transform-factor pairs that can optimally characterize the original time-resolved signal will be a challenge in the future investigation.

^{-1}μM

^{-1}, the lifetime of about 0.56 ns and the quantum efficiency of 0.016 at the peak excitation wavelength of 778 nm) and Cy5.5 (with the extinction coefficient of about 0.019 mm

^{-1}μM

^{-1}, the lifetime of about 1 ns and the quantum efficiency of 0.23 at the peak excitation wavelength of 670 nm). Nevertheless, with regard to these two dyes, the fluorophore concentrations corresponding to the chosen fluorescent yields of the targets are all on the orders of 1μM and 100

*n*M, respectively, as preferably expected

*in vivo*.

*i.e.*, Eq. (17), is critical to the reconstruction fidelity. Normally, this would be attainable through regularizing ill-posedness of the inverse issue and incorporating

*a priori*knowledge. It is also argued that two-component reconstruction is target-dependent and would be improved for targets of large-size and low-contrast.

*in vivo*experimental validations of the methodology in the on-going work.

## References and links

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11. | V. Ntziachristos, C. Bremer, E.E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Molecular Imaging |

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

13. | 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. |

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

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

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

17. | A.D. Klose, V. Ntziahristos, and A.H. Hielschler, “The inverse source problem based on the reative trabsfer equation in optical molecular imaging,” J. Comput. Phys. |

18. | S.R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

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. | R. Model, M. Orlt, and M. Walzel, “Reconstruction algorithm for near-infrared imaging in turbid media by means of time-domain data,” J. Opt. Soc. Am. A |

21. | M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. |

22. | F. Gao, P. Poulet, and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from a three-dimensional model of time-resolved optical tomography,” App. Opt |

23. | 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. |

24. | J.C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D.T. Delpy, S.R. Arridge, J.H. Meek, and J.S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. |

25. | F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by time-domain detection: methodology and phantom validation,” Phys. Med. Biol. |

26. | Huijuan Zhao, Gao Feng, Yukari Tanikawa, Kazuhiro Homma, and Yukio Yamada, “Time-resolved optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. |

27. | 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. |

28. | W.G. Egan and T.W. Hilgeman, |

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

30. | F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image and Vision Computing |

31. | F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Thromberg, and C. Depeursinge, “ |

32. | 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 |

**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: May 17, 2006

Revised Manuscript: July 24, 2006

Manuscript Accepted: July 24, 2006

Published: August 7, 2006

**Virtual Issues**

Vol. 1, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Feng Gao, Huijuan Zhao, Yukari Tanikawa, and Yukio Yamada, "A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomography," Opt. Express **14**, 7109-7124 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7109

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### References

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