## A self-normalized, full time-resolved method for fluorescence diffuse optical tomography

Optics Express, Vol. 16, Issue 17, pp. 13104-13121 (2008)

http://dx.doi.org/10.1364/OE.16.013104

Acrobat PDF (697 KB)

### Abstract

A full time-resolved scheme that has been previously applied in diffuse optical tomography is extended to time-domain fluorescence diffuse optical tomography regime, based on a finite-element-finite-time-difference photon diffusion modeling and a Newton-Raphson inversion framework. The merits of using full time-resolved data are twofold: it helps evaluate the intrinsic performance of time-domain mode for improvement of image quality and set up a valuable reference to the assessment of computationally efficient featured-data-based algorithms, and provides a self-normalized implementation to preclude the necessity of the scaling-factor calibration and spectroscopic-feature assessments of the system as well as to overcome the adversity of system instability. We validate the proposed methodology using simulated data, and evaluate its performances of simultaneous recovery of the fluorescent yield and lifetime as well as its superiority to the featured-data one in the fidelity of image reconstruction.

© 2008 Optical Society of America

## 1. Introduction

8. S. Lam, F. Lesage, and X. Intes X, “Time-domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express **13**, 2263–2275 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2263. [CrossRef] [PubMed]

9. F. Gao, H. J. Zhao, Y. Tanikawa, and Y. 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/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]

7. A. T. N. Kumar, S. B. Raymond, G. Boverman, D. A. Boas, and B. J. Bacskai, “Time-resolved fluorescence tomography of turbid media based on lifetime contrast,” Opt. Express **14**, 12255–12270 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12255. [CrossRef] [PubMed]

17. 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]

17. 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]

9. F. Gao, H. J. Zhao, Y. Tanikawa, and Y. 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/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]

## 2. Forward model - a finite-element-finite-difference solution to coupled time-domain diffusion equations

*x*and

*m*denote the excitation and emission wavelengths, respectively. Then the propagation of both excitation and emission light in a heterogeneous, fluorescent turbid medium Ω, as externally excited by an ideal ultra-short point source (

*i.e.*δ -shaped in space and time), can be modeled by the coupled time-domain diffusion equations as follows [18

18. T. J. Farrell and M. S. Patterson, “Diffusion modeling of fluorescence in tissue,” in Handbook of Biomedical Fluorescence, Mycek MA and Pogue BW eds., Marcel Dekker, New York (2003). [CrossRef]

_{ν}(

**r**,

**r**

*,*

_{s}*t*) (ν∈[

*x*,

*m*]) (

*t*: time,

**r**: position in Ω) is the temporally- and spatially-varying photon density corresponding to a δ -shaped source at position

**r**

_{s}; the optical properties involved are the absorption coefficient μ

_{aν}(

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

_{sν}(

**r**) and the diffusion coefficient κ

_{ν}(

**r**,

*t*)=

*c*/[3μ′

_{sν}(

**r**)] at the two wavelengths, respectively; the fluorescent parameters are the yield η(

**r**) (product of the quantum efficiency and the absorption coefficient of the fluorophore) and the lifetime τ(

**r**);

*e*(τ,

*t*)=

*e*

^{-t/τ}

*U*(

*t*) with

*U*(

*t*) being a unit-step function; the operator ⊗ denotes the temporal convolution. We employ uniquely the Robin boundary condition for the above coupled equations. The measurable temporally-varying flux density,

*i.e.*, the temporal point spread function (TPSF), at the detection site ξ

_{d}(

*d*=1,2,…,D) on the boundary ∂Ω for the source site ζ

_{s}(

*s*=1,2,…,S), can be calculated by the Fick’s law under the Robin boundary condition [19

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

*R*≈0.53 is the internal reflection coefficient for air-tissue interface [20]. For finite-element-method (FEM) solution to Eq.(1), ϕ

_{f}_{ν}(

**r**,

*t*) (

**r**

*is omitted for a brevity) is firstly approximated by a piecewise-polynomial function*

_{s}**u**(

**r**)=[

*u*

_{1}(

**r**),…,

*u*(

_{N}**r**)]

^{T}and

**Φ**

_{ν}(

*t*)=[Φ

_{ν}(1,

*t*),…,Φ

_{ν}(

*N*,

*t*)]

^{T}are vectors representing the shape functions and temporally-varying photon density at the

*N*nodes of the FEM mesh. Then, a temporally-recursion matrix equation can be obtained below by employing the standard Galerkin-FEM procedure following by a full implicit time-domain-finite-difference (TDFD) scheme [17

17. 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. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite approach for modeling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

*t*:

_{f}**Φ**

_{ν}(

*i*)=

**Φ**

_{ν}(

*i*Δ

*t*),

_{f}*i*=-1,0,1,…

**A**

_{ν,}

**B**and

**C**are matrices of

*N*×

*N*with the same expression as given in [21

21. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite approach for modeling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

*i.e.*

**Q**

_{ν}(

*i*) differs in form

**Φ**′

_{x}(

*i*)=[Φ′

*(1,*

_{x}*i*),…,Φ′

*(*

_{x}*N*,

*i*)]

*with*

^{T}*n*) and τ(

*n*) being the fluorescent yield and lifetime at the

*n*-th node, respectively, and

*E*(

*n*,

*i*)=

*e*(τ(

*n*),

*i*Δ

*t*); δ(

_{f}*i*) is a Kronecker δ -function. Since the matrix (

**A**

_{ν}+

**B**)+

**C**/Δ

*t*is constant throughout the TDFD recursion in Eq. (4), the Choleski decomposition is needed only once during the solution process [21

_{f}21. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite approach for modeling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

*R*=15 mm and optical and fluorescent properties of μ

^{(B)}

_{ax,m}=0.035 mm

^{-1}μ′

^{(B)}

_{sx,m}=1.0 mm

^{-1}, η

^{(B)}=0.001 mm

^{-1}and τ

^{(B)}=500 ps ((B) denotes the background). The geometry and parameters are chosen for the potential applications of FMT in small animals as well as the status of the currently favorable red-shifted or NIR fluorescent probes [3

3. V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. **8**, 757–760 (2002). [CrossRef] [PubMed]

8. S. Lam, F. Lesage, and X. Intes X, “Time-domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express **13**, 2263–2275 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2263. [CrossRef] [PubMed]

22. S. Achilefu, P. R. Dorshow, J. E. Bugaj, and R. Rajapopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. **35**, 479–485 (2000). [CrossRef] [PubMed]

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

*r*=3 mm and the same optical properties as the background is embedded at the center. The domain is excited by a δ- shaped laser pulse at a polar angle of θ=0° and the excitation re-emission and fluorescent emission are temporally measured at θ=180°, respectively. The FEM mesh contains 2400 triangle elements that join at 1261 nodes. Figure 1(a) shows the geometrical setup and the FEM mesh. In Case 1, the fluorescent inclusion has a fixed yield of η=0.002 mm

^{-1}but different lifetimes of τ=560, 1000, 2800 ps, respectively, covering the range of the commonly-used fluorophore agents, such as Indocyanine Green (ICG) and Cy-series (Cy5.5, Cy3-B)

*etc.*[8

8. S. Lam, F. Lesage, and X. Intes X, “Time-domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express **13**, 2263–2275 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2263. [CrossRef] [PubMed]

*t*=10, 20, 40, 80 ps. The results show that the relative error of the maximum TPSF points for the two successive steps of the TDFD-interval: Δ

_{f}*t*=10 ps and 20 ps, is smaller than 10

_{f}^{-3}. It is requested that, for assuring an adequate accuracy of the forward calculation while avoiding an excess computational cost, a critical TDFD-interval,

*e.g.*, 20 ps for the case, should be suitably decided.

## 3. Inverse model - a self-normalized, full time-resolved scheme for simultaneous reconstruction of fluorescent yield and lifetime

**F**

_{m}[

**p**(

**r**)]=[

*F*(ξ

_{m}_{1},ζ

_{1},

*t*

_{1},

**p**(

**r**)),…,

*F*(ξ

_{m}_{D},ζ

_{S},

*t*,

_{I}**p**(

**r**))]

^{T}symbolize the forward operator that maps the image space

**p**(

**r**)=[η(

**r**), τ(

**r**)]

^{T}into the data space

**Γ**

*=[Γ*

_{m}*(ξ*

_{m}_{1},ζ

_{1},

*t*

_{1}),…,Γ

*(ξ*

_{m}*,ζ*

_{D}*,*

_{S}*t*)]

_{I}^{T}, with

*D*,

*S*and

*I*being the numbers of the sources, detectors and temporal samples, respectively, and subscript

*m*emphasizing the emission wavelength. Due to the nonlinear dependency of the measurements on τ(

**r**), the inverse model for the simultaneous recovery of both the yield and lifetime can be in general addressed by the Newton-Raphson scheme, based on the Taylor series expansion of

**F**

_{m}[

**p**(

**r**)]. This forms the outer-loop of an iterative procedure

**J**(

**p**) is the Fréchet derivatives of

**F**

_{m}[

**p**(

**r**)] with respect to η(

**r**) and τ(

**r**). Its discrete formula,

*i.e.*, the Jacobian matrix, can be efficiently calculated from a perturbation approach to the emission diffusion equation, firstly leading to an integral expression as follows, associating the parameter perturbations, δη(

**r**) and δτ(

**r**) with the incurred measurement change δ

**Γ**

_{m},

*f*

_{η}(τ,

*t*)=

*e*(τ,

*t*)/τ and

*f*

_{τ}(τ,η,

*t*)=

*e*(τ,

*t*)(

*t*-τ)η/τ

^{3};

*g*(

_{m}**r**,

**r**′,

*t*) is the Green’s function associated with the emission diffusion equation. Similarly, the fluorescent yield and lifetime distributions can be decomposed over the shape function basis of the finite elements: δη(

**r**)=δ

**η**

^{T}

**u**(

**r**) and δτ(

**r**)=δ

**τ**

^{T}

**u**(

**r**), where δ

**η**=[δη(1),δη(2),…,δη(

*N*)]

^{T}and δ

**τ**=[δτ(1),δτ(2),…,δτ(

*N*)]

*with δη(*

^{T}*n*) and δτ(

*n*) being the yield and lifetime perturbations at the n-th node, respectively. Then, the n-th entries of the η- and τ- regarding Jacobian matrices for source site ζ

*, detector site ξ*

_{d}*and time*

_{s}*t*,

_{i}**J**

_{η}(ξ

*,ζ*

_{d}*,*

_{s}*t*) and

_{i}**J**

_{τ}(ξ

*,ζ*

_{d}*,*

_{s}*t*), are given, respectively, upon the spatial discretization

_{i}_{n→j}denotes the j-th element associated with the n-th node (i.e., the element having the n-th node as its common vertex),

*V*(Ω

_{n→j}) is the volume (or area) of element Ω

_{n→j}, and

_{n→j}, which is approximated by the average of the values at the nodes of Ω

_{n→j}. The factor

*W*=3 and 4 for a triangle and a tetrahedron, respectively. The twofold temporal convolutions involved in the above expressions are carried-out in the Fast Fourier Transform that is more time-saving but storage-costly than the direct linear convolution.

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

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

*i.e.*, those that belong to the same element as the node of interest) at the end of each out-loop iteration, as it has been done in our previous studies [9

9. F. Gao, H. J. Zhao, Y. Tanikawa, and Y. 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/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]

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

25. H. J. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. **12**, Art. No. 062107 (2007). [CrossRef] [PubMed]

### 3.1 Time-bin νersion

*i.e.*, the so-called time-bin, which is determined either by the bandwidth of the instrumental impulse response in an analog mode, or by the jitter performance of the output pulses in a digital one, such as the transit-time-spread (TTS) of the PMT in the TCSPC detection, or the gate-width in a boxcar mode, such as a gated ICCD camera, and accounts for the physical temporal resolution of the time-resolved measuring systems [26

26. W. Becker, *Advanced time-correlated single photon counting techniques* (Springer-Verlag, Berlin2005). [CrossRef]

*T*, the data-type is written as

*t*=

_{k}*T*[rand(

*k*)-0.5] with rand(

*k*) being a random variable uniformly distributed in [0,1] and

*K*is an associate number with the photon counting. Accordingly, the Jacobian matrices with regard to the time-bin data-type become

_{m}(ζ

*,ξ*

_{d}*,*

_{s}*t*) spaced at an sampling-interval of Δ

_{i}*t*,

_{r}*i.e.*,

*t*=

_{i}*i*Δ

*t*,

_{r}*i*=

*I*

_{1},

*I*

_{1}+1, ⋯,

*I*

_{2}, where integers

*I*

_{1}and

*I*

_{2}usually vary with the sourcedetector pairs to cover the most effective range of the individual TPSFs, and Δ

*t*, is in general determined by either the electronic resolution of a TCSPC system or the delay-interval of a gated ICCD camera.

_{r}### 3.2 Normalization

27. D. A. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express **8**, 253–270 (2001). [CrossRef]

25. H. J. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. **12**, Art. No. 062107 (2007). [CrossRef] [PubMed]

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

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

*i.e.*,

*F̅*(ξ

_{m}*,ζ*

_{d}*,*

_{s}*t*,

_{i}**η**

_{k},

**τ**

_{k}) is the model-calculation of the temporally binned flux at the emission wavelength;

*F*

^{(E)}

^{x}(ξ

*,ζ*

_{d}*) is the model-calculation of the integrated intensity at the excitation wavelength; Θ*

_{s}*and Θ*

_{x}*are the scaling factors of the system at the excitation and emission wavelengths, respectively. The more effective way is to construct a self-normalization that normalizes the temporally binned flux*

_{m}_{m}(ξ

*,ζ*

_{d}*,*

_{s}*t*) to its integrated intensity:

*F*

^{(E)}

_{m}(ξ

*,ζ*

_{d}*,*

_{s}**η**

*) is the model-calculations of the integrated intensity at the emission wavelength, respectively;*

_{k}12. J. Lee and E. M. Sevick-Muraca, “Three-dimensional fluorescence enhanced optical tomography using referenced frequency-domain photon migration measurements at emission and excitation wavelengths,” J. Opt. Soc. Am. A **19**, 759–771 (2002). [CrossRef]

13. R. Roy, A. B. Thompson, A. Godavarty, and E. M. Sevick-Muraca, “Tomographic fluorescence imaging in tissue phantom: A novel reconstruction algorithm and imaging geometry,” IEEE Trans. Med. Imaging **24**, 137–154 (2005). [CrossRef] [PubMed]

*E*(ξ

_{m}*,*

_{d}**r**) is the emission Green’s function and Φ

*(*

_{x}**r**,ζ

*) the excitation photon density.*

_{s}5. S. R. Cherry, “In vivo molecular and genomic imaging: new challenges for imaging physics,” Phys. Med. Biol. **49**, R13–R48 (2004). [CrossRef] [PubMed]

30. 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. **49**, 1117–1130 (2004). [CrossRef] [PubMed]

31. H. J. 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. **43**, 1905–1916 (2005). [CrossRef]

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

**14**, 7109–7124 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]

## 4. Methodology evaluations

16. M. Brambilla, L. Spinelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Time-resolved scanning system for double reflectance and transmittance fluorescence imaging of diffusive media,” Rev. Sci. Instrum. **79**, 013103 (2008). [CrossRef] [PubMed]

26. W. Becker, *Advanced time-correlated single photon counting techniques* (Springer-Verlag, Berlin2005). [CrossRef]

32. F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. **71**, 256–265 (2000). [CrossRef]

2. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas DA, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. **42**, 3081–3094 (2003). [CrossRef] [PubMed]

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

15. S. V. Patwardhan and J. P. Culver, “Quantitative diffuse optical tomography for small animals using an unltrafast gated image intensifier,” J. Biomed. Opt. **13**, 011009 (2008). [CrossRef] [PubMed]

*T*=180 ps for the TCSPC or

*T*≥500 ps for the gated ICCD would be reasonable. To facilitate the quantification, two circular targets with the same optical properties as those of the background are aligned along the Y-axis, symmetric to the origin. They have different radius, r1 and r2, as well as fluorescent parameter sets, (η

_{1},τ

_{1}) and (η

_{2},τ

_{2}), respectively. For any reconstructed images, their Y-profiles,

*i.e.*, the cut lines along the Y-axis with notations

*y*) and

*y*) for the yield and lifetime, respectively, are extracted to evaluate the measures. We firstly investigate the intrinsic performance of the methodology with noiseless data, and then check its noise-robustness using a noise model in accordance with the physics of the fluorescence generation, propagation and detection. To validate the robustness of the method on the numerical error of the forward model, a fine mesh with 3750 elements and 1951 nodes is used in the data-generator, as compared with a coarse mesh for the inversion-solver that contains 2400 elements and 1261 nodes. Also, to assure an adequately high signal-to-noise ratio (SNR), we use the temporal samples ranging from 10% of the maximum intensity on the rising edge to 10% of the maximum intensity on the falling edge for each of the source-detector pairs. It is worth notice that some researchers have addressed the optimization of temporal measurements using a singular-value-decomposition analysis and shown a redundancy-reduction mode by obtaining an optimal result from the rise and the peak portions of the TPSFs [7

7. A. T. N. Kumar, S. B. Raymond, G. Boverman, D. A. Boas, and B. J. Bacskai, “Time-resolved fluorescence tomography of turbid media based on lifetime contrast,” Opt. Express **14**, 12255–12270 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12255. [CrossRef] [PubMed]

### 4.1 Spatial-resolution

*R*

_{χ}(χ∈[

*r*

_{1}=

*r*

_{2}=2 mm and fluorescent contrast to the background of 5:1 and 3:1 for the yield and lifetime, respectively,

*i.e.*, η

_{1}=η

_{2}=0.005 mm

^{-1}and τ

_{1}=τ

_{2}=1500 ps. To investigate the effects of temporal features of the detectors on the reconstruction, a few of different time-bin widths and sampling intervals are used in the reconstruction. Figure 3 shows the e

*R*

_{χ}as functions of the CCS for different sampling-intervals and different bin-widths, respectively. To better demonstrate the superiority of the time-resolved method to the featured-data one, the results obtained with the GPST algorithm are presented for a comparison. Figure 4 illustrates the reconstructed images using the proposed scheme for a fixed sampling-interval of

*t*=20 ps and a series of the bin-widths:

_{r}*T*=20, 180, 500 and 1000

*ps*, while Figure 5 shows the reconstructed η - and τ -images using the proposed method for a fixed bin-width of

*T*=180

*ps*and different sampling-intervals of

*t*=20, 100, and 180

_{r}*ps*, both at a fixed

*CCS*=11 mm. It is noted from the results that the sampling-interval greatly affects spatial resolution of the reconstruction, and at a small sampling,

*e.g.*,

*t*=20

_{f}*ps*, the moderate increase of the bin-width (<500

*ps*) has no evidently adverse influence on the image quality.

### 4.2 Quantitativeness

*Q*

_{χ}for the evaluation

*i.e.*, (η

*,τ*

_{i}*)=(*

_{i}*j*0.001 mm

^{-1},

*j*500 ps),

*i*=1,2;

*j*=2, 3,…,8. Figure 6 calculates the reconstructed fluorescent contrast

*Q*

_{χ}as a function of the target fluorescent contrast (

*i.e.*, the ratio of the maximum to the minimum in the target Y-profile) for different sampling-intervals and different bin-widths, respectively. For contrasting between the two schemes, the results from the GPST method are also included. The similar conclusion can be obtained from the results: the yield and (low-contrasted) lifetime reconstructions with high quantitativeness can be attained at a small sampling interval of

*t*=20

_{f}*ps*, and are insensitive to an increase in the bin-width up to 500

*ps*.

### 4.3 Size-contrast

*S*

_{χ}as a measure of the fidelity of the methodology to distinguish the difference in the target size between the two targets, is defined as

^{(min)}=min[χ(

*y*)] is the minimum value of the Y-profile;

*y*

^{(l)}

_{i}and

*y*

^{(u)}

_{i}(

*i*=1,2) are the

*y*-coordinates of the half-maximum of the target 1 and 2, respectively. Again, the same scenario as before is used with a fixed d

*CCS*=15 mm for the two circular targets and fluorescent contrast of 5:1 and 3:1 for the target yield and lifetime, respectively. Figure 7 gives the

*S*

_{χ}as a function of the target size-contrast (defined as the ratio of the diameter of the target 1 to that of the target 2), ranging from 1 to 3,

*i.e.*,

*r*

_{1,2}=2±0.25

*j*mm,

*j*=0,1,…,5. Similarly, the calculations are made for different sampling-intervals and different bin-widths and compared to the GPST results. Again, we have the similar observation as the above that the measure is adversely affected by the increase in the sampling-interval and nearly unchanged with the bin-width under 500

*ps*.

### 4.4 Gray-resolution

*G*

_{χ}, is used to quantify the ability for the method to reconstruct the gray difference

^{(max)}

_{1}, χ

^{(max)}

_{2}and χ

^{(min)}are afore-defined. We quantify the performance by calculating

*G*

_{χ}as a function of the target grayscale difference at a fixed baseline of η=0.005 mm

^{-1}and τ=1500 ps,

*i.e.*, the mean fluorescent parameters of the two targets with the radius of

*r*

_{1}=

*r*

_{2}=2 mm and

*CCS*=15 mm, as shown in Fig. 8. In addition, we also evaluate

*G*

_{χ}as a function of the target η-baseline at a fixed target lifetime baseline of τ=1500 ps, a target grayscale difference of 30% and a target radius of

*r*

_{1}=

*r*

_{2}=2 mm, and as a function of the target radius at a fixed target baseline of η=0.005 mm

^{-1}and τ=1500 ps and a target grayscale difference of 30% (the results are not given for the brevity). The evaluations are contrasted for different sampling-intervals and different bin-widths, respectively, as well as compared with the GPST results. For better demonstrating the effect of the sampling-interval selection on the reconstruction fidelity of the grayscale difference, Figure 9 illustrates the reconstructed images of the above first scenario with

*t*=20, 100, and 180 ps, respectively, at a fixed bin width of

_{r}*T*=180 ps. It is seen that from Fig. 8, for the yield reconstruction an optimal result is achieved at a sampling-interval of

*t*=20

_{f}*ps*, as compared to the GPST method that over-estimates the measure, and to the proposed one with a larger time-interval that under-estimates the true value. As to the lifetime reconstruction, the GPST method gives an over-estimation while the proposed one underestimation with the worst result generated at

*t*=20

_{f}*ps*. The observation is further manifested in Fig. 9.

### 4.5 Noise-robustness

*t*in a given measurement (integration) time [26

26. W. Becker, *Advanced time-correlated single photon counting techniques* (Springer-Verlag, Berlin2005). [CrossRef]

*N*is the number of the detected photons corresponding to min[Γ

_{pm}*(ξ*

_{m}*,ζ*

_{d}*,*

_{s}*t*)]. We introduce the mean value of

*SNR*(ξ

*,ζ*

_{d}*,*

_{s}*t*) over the time range, to depict the overall noise-level in the measured TPSFs.

*T*=20, 180, and 500 ps, respectively, at

*t*=20 ps, where a standard target settings with η - and τ -contrasts of 5:1 and 3:1, respectively, a radius of

_{r}*r*

_{1}=

*r*

_{2}=2 mm and a CCS of 15 mm are used. An overall improvement in the image quality can be evidently observed at

*T*=500

*ps*

## 5. Discussions

*e.g.*, the GPST scheme, perhaps, the biggest gain with using full time-resolved data lies on the significant improvement in the spatial resolution of both the yield and lifetime images. This is achieved for both the yield and lifetime reconstructions at a sufficiently narrow sampling-interval,

*e.g.*,

*t*=20 ps, as clearly seen in Fig. 3. The observations are further enforced with the reconstructed images in Fig. 4 and Fig. 5, where the resultant images exhibit a quality consistency for

_{r}*T*≤500 ps and an obvious quality degradation with the increasing sampling-interval. According to Fig. 6, the proposed full time-resolved scheme significantly improves quantitativeness of the yield reconstruction but evidently degrades the quantitation in the lifetime image for a target-to-background contrast higher than 4:1. The latter disadvantage is primarily ascribed to the increasingly saturating sensitivity of the detected signals to the long-lifetime fluorescent variation. As illustrated in Fig. 7, the size-contrast reconstructions with the proposed method is slightly under-estimated for the yield reconstruction and slightly over-estimated for the lifetime one, whereas the GPST method presents an severely over-estimated reconstruction for the yield and a slight over-estimation for the lifetime, in terms of the measure

*S*

_{χ}. Similarly, an optimal reconstruction fidelity of the target size contrast is found at a small time-interval of

*t*=20

_{r}*ps*. The evaluations on the grayscale performance of the method show that the reconstruction with the full time-resolved method is under-estimated for both the yield and lifetime, contrary to those with the GPST, as shown in Fig. 8. Further evaluations on this performance as functions of the target contrast and size, respectively, show that with the increase in the yield contrast and target radius, the grayscale reconstruction of the yield gets successively improved while the reconstructed grayscale of the lifetime degrades slightly. Roughly, the results conform to our previous observations in DOT [33

33. F. Gao, H. J. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography - a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. **37**, 1287–1304 (2005). [CrossRef]

*et. al.*have investigated the influence of local variations in the optical properties on the accuracy of the fluorescence reconstruction and demonstrated that, the reasonable quantification is relatively insensitive to a degree of background heterogeneities, with the excitation-normalized Born approach [34

34. 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]

35. F. Gao, P. Poulet, and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from full three-dimensional model of time-resolved optical tomography,” Appl. Opt. **39**, 5898–5910 (2000). [CrossRef]

**20**, 299–309 (1993). [CrossRef] [PubMed]

25. H. J. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. **12**, Art. No. 062107 (2007). [CrossRef] [PubMed]

**14**, 7109–7124 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]

*x*=0 mm,

*y*=5.5 mm) a circular target with a radius of

*r*=3 mm and η - and τ - contrasts of 5:1 and 3:1, respectively. According to the results a margin of the time origin within ±80 ps is required for a reasonable output, which is somewhat rigorous in practice. Normally, the solution to this issue is strongly dependent on the availability of the calibrating tools that directly measure the absolute time-origins of the system channels with an acceptable accuracy. Although some effective efforts have been made for this direct strategy [36

36. E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum. **71**, 3415–3427 (2000). [CrossRef]

*i.e.*, the difference in size is often converted in part to difference in contrast, and vice versa. Therefore, the size-contrast and gray-resolution measures introduced in this paper are necessarily required for the completeness and objectiveness of the methodology evaluations.

## 6. Conclusions

- Compared to the Born formulation where the ratio of emission data to excitation one is employed for the elimination of the system scaling, the self-normalization scheme possesses additional advantages of enhancing the robustness to system fluctuations and exempting spectroscopic characterization of the system, whereas the performance degradation incurred due the loss in the intensity information is unconspicuous;
- A noise-robust image reconstruction might be attained with reasonable fidelity by properly increasing the bin-width,
*e.g.*, up to 500 ps, while remaining the sampling-interval enough narrow,*e.g.*, down to 25 ps. This feature technically allows for the application of a state of the art ICCD for dense spatial sampling, and offers a favorable way of improving the SNR.

## Acknowledgments

## References and Links

1. | E. M. Sevick-Muraca, J. P. Houston, and M. Gurfinkel, “Fluorescence-enhanced, near infrared diagnostic imaging with contrast agent,” Curr. Opin. Chem. Biol. |

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

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

4. | V. Ntziachiristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. |

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

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

7. | A. T. N. Kumar, S. B. Raymond, G. Boverman, D. A. Boas, and B. J. Bacskai, “Time-resolved fluorescence tomography of turbid media based on lifetime contrast,” Opt. Express |

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

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

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

11. | A. D. Klose, V. Ntziachristos, and A. D. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. |

12. | J. Lee and E. M. Sevick-Muraca, “Three-dimensional fluorescence enhanced optical tomography using referenced frequency-domain photon migration measurements at emission and excitation wavelengths,” J. Opt. Soc. Am. A |

13. | R. Roy, A. B. Thompson, A. Godavarty, and E. M. Sevick-Muraca, “Tomographic fluorescence imaging in tissue phantom: A novel reconstruction algorithm and imaging geometry,” IEEE Trans. Med. Imaging |

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

15. | S. V. Patwardhan and J. P. Culver, “Quantitative diffuse optical tomography for small animals using an unltrafast gated image intensifier,” J. Biomed. Opt. |

16. | M. Brambilla, L. Spinelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Time-resolved scanning system for double reflectance and transmittance fluorescence imaging of diffusive media,” Rev. Sci. Instrum. |

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

18. | T. J. Farrell and M. S. Patterson, “Diffusion modeling of fluorescence in tissue,” in Handbook of Biomedical Fluorescence, Mycek MA and Pogue BW eds., Marcel Dekker, New York (2003). [CrossRef] |

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

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

21. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite approach for modeling photon transport in tissue,” Med. Phys. |

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

23. | K. Licha, “Contrast agents for optical imaging,” Topics in Current Chemistry |

24. | O. C. Zienkiewicz and R. L. Taylor, |

25. | H. J. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. |

26. | W. Becker, |

27. | D. A. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express |

28. | S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in threedimensional Baysian optical diffusion tomography,” J. Opt. Soc. Am. A |

29. | T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and A. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. |

30. | 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. |

31. | H. J. 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. |

32. | F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. |

33. | F. Gao, H. J. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography - a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. |

34. | 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 |

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

36. | E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

(170.5280) Medical optics and biotechnology : Photon migration

(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 1, 2008

Revised Manuscript: August 5, 2008

Manuscript Accepted: August 6, 2008

Published: August 12, 2008

**Virtual Issues**

Vol. 3, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Feng Gao, Huijuan Zhao, Limin Zhang, Yukari Tanikawa, Andhi Marjono, and Yukio Yamada, "A self-normalized, full time-resolved method for
fluorescence diffuse optical tomography," Opt. Express **16**, 13104-13121 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-17-13104

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### References

- E. M. Sevick-Muraca, J. P. Houston, and M. Gurfinkel, "Fluorescence-enhanced, near infrared diagnostic imaging with contrast agent," Curr. Opin. Chem. Biol. 6, 642-650 (2002). [CrossRef] [PubMed]
- A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas DA, and R. P. Millane, "Fluorescence optical diffusion tomography," Appl. Opt. 42, 3081-3094 (2003). [CrossRef] [PubMed]
- V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, "Fluorescence molecular tomography resolves protease activity in vivo," Nat. Med. 8, 757-760 (2002). [CrossRef] [PubMed]
- V. Ntziachiristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotech. 23, 313-320 (2005) [CrossRef]
- S. R. Cherry, "In vivo molecular and genomic imaging: new challenges for imaging physics," Phys. Med. Biol. 49, R13-48 (2004). [CrossRef] [PubMed]
- X. Cong and G. Wang, "A finite-element-based reconstruction method for 3D fluorescence tomography," Opt. Express 13, 9847-9857 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-24-9847. [CrossRef] [PubMed]
- A. T. N. Kumar, S. B. Raymond, G. Boverman, D. A. Boas, and B. J. Bacskai, "Time-resolved fluorescence tomography of turbid media based on lifetime contrast," Opt. Express 14, 12255-12270 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12255. [CrossRef] [PubMed]
- S. Lam, F. Lesage, and X. Intes X, "Time-domain fluorescent diffuse optical tomography: analytical expressions," Opt. Express 13, 2263-2275 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2263. [CrossRef] [PubMed]
- F. Gao, H. J. Zhao, Y. Tanikawa, and Y. 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/abstract.cfm?URI=oe-14-16-7109. [CrossRef] [PubMed]
- H. Jiang, "Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulation," Appl. Opt. 37, 5337-5343 (1998). [CrossRef]
- A. D. Klose, V. Ntziachristos, and A. D. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005). [CrossRef]
- J. Lee and E. M. Sevick-Muraca, "Three-dimensional fluorescence enhanced optical tomography using referenced frequency-domain photon migration measurements at emission and excitation wavelengths," J. Opt. Soc. Am. A 19, 759-771 (2002). [CrossRef]
- R. Roy, A. B. Thompson, A. Godavarty, and E. M. Sevick-Muraca, "Tomographic fluorescence imaging in tissue phantom: A novel reconstruction algorithm and imaging geometry," IEEE Trans. Med. Imaging 24, 137-154 (2005). [CrossRef] [PubMed]
- V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, "Fluorescence lifetime imaging by using time-gated data," Appl. Opt. 46, 7384-7391 (2007). [CrossRef] [PubMed]
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- W. Becker, Advanced time-correlated single photon counting techniques (Springer-Verlag, Berlin 2005). [CrossRef]
- D. A. Boas, T. Gaudette, and S. R. Arridge, "Simultaneous imaging and optode calibration with diffuse optical tomography," Opt. Express 8, 253-270 (2001). [CrossRef]
- S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, "Source-detector calibration in three-dimensional Baysian optical diffusion tomography," J. Opt. Soc. Am. A 19, 1983-1993 (2002). [CrossRef]
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- 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. 49, 1117-1130 (2004). [CrossRef] [PubMed]
- H. J. 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. 43, 1905-1916 (2005). [CrossRef]
- F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy, "A 32-channel time-resolved instrument for medical optical tomography," Rev. Sci. Instrum. 71, 256-265 (2000). [CrossRef]
- F. Gao, H. J. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, "Influences of target size and contrast on near infrared diffuse optical tomography - a comparison between featured-data and full time-resolved schemes," Opt. Quantum Electron. 37, 1287-1304 (2005). [CrossRef]
- 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]
- F. Gao, P. Poulet, and Y. Yamada, "Simultaneous mapping of absorption and scattering coefficients from full three-dimensional model of time-resolved optical tomography," Appl. Opt. 39, 5898-5910 (2000). [CrossRef]
- E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, "Calibration techniques and datatype extraction for time-resolved optical tomography," Rev. Sci. Instrum. 71, 3415-3427 (2000). [CrossRef]

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