## Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm

Optics Express, Vol. 15, Issue 26, pp. 18300-18317 (2007)

http://dx.doi.org/10.1364/OE.15.018300

Acrobat PDF (642 KB)

### Abstract

With the development of *in-vivo* free-space fluorescence molecular imaging and multi-modality imaging for small animals, there is a need for new reconstruction methods for real animal-shape models with a large dataset. In this paper we are reporting a novel hybrid adaptive finite element algorithm for fluorescence tomography reconstruction, based on a linear scheme. Two different inversion strategies (Conjugate Gradient and Landweber iterations) are separately applied to the first mesh level and the succeeding levels. The new algorithm was validated by numerical simulations of a 3-D mouse atlas, based on the latest free-space setup of fluorescence tomography with 360° geometry projections. The reconstructed results suggest that we are able to achieve high computational efficiency and spatial resolution for models with irregular shape and inhomogeneous optical properties.

© 2007 Optical Society of America

## 1. Introduction

*in-vivo*small animal imaging because of its ability to resolve 3D spatial distributions of fluorescence probes associated with molecular and cellular functions. Many efforts have been made to develop new probes, photon migration models, imaging systems, and the corresponding reconstruction methods [1

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

2. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. **8**, 1–33 (2006). [CrossRef] [PubMed]

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

4. J. Chang, H. L. Graber, and R. L. Barbour, “Luminescence optical tomography of dense scattering media,” JOSA A **14**, 288–299 (1997). [CrossRef] [PubMed]

5. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. **30**, 901–911 (2003). [CrossRef] [PubMed]

6. N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. **32**, 382–384 (2007). [CrossRef] [PubMed]

7. H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. N. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt. **46**, 3617–3627 (2007). [CrossRef] [PubMed]

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

2. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. **8**, 1–33 (2006). [CrossRef] [PubMed]

2. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. **8**, 1–33 (2006). [CrossRef] [PubMed]

8. X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. **30**, 861–869 (2003). [CrossRef] [PubMed]

9. A. Joshi, W. Bangerth, and E.M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

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

11. D. Wang, X. Song, and J. Bai, “A novel adaptive mesh based algorithm for fluorescence molecular tomography using analytical solution,” Opt. Express **15**, 9722–9730 (2007). [CrossRef] [PubMed]

9. A. Joshi, W. Bangerth, and E.M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

11. D. Wang, X. Song, and J. Bai, “A novel adaptive mesh based algorithm for fluorescence molecular tomography using analytical solution,” Opt. Express **15**, 9722–9730 (2007). [CrossRef] [PubMed]

*a priori*information derived from a previous reconstruction procedure. Considering this aspect, the first reconstruction procedure is different from the succeeding ones, in that it does not have

*a priori*information from a previous procedure. However, each of the four papers mentioned [8

8. X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. **30**, 861–869 (2003). [CrossRef] [PubMed]

11. D. Wang, X. Song, and J. Bai, “A novel adaptive mesh based algorithm for fluorescence molecular tomography using analytical solution,” Opt. Express **15**, 9722–9730 (2007). [CrossRef] [PubMed]

8. X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. **30**, 861–869 (2003). [CrossRef] [PubMed]

**15**, 9722–9730 (2007). [CrossRef] [PubMed]

12. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging **23**, 492–500 (2004). [CrossRef] [PubMed]

14. D. Hyde, A. Soubret, J. Dunham, T. Lasser, E. Miller, D. Brooks, and V. Ntziachristos, “Analysis of reconstructions in full view fluorescence molecular tomography,” Proc. SPIE **6498**, 649803 (2007). [CrossRef]

15. X. Li, B. Chance, and A. G. Yodh, “Fluorescent heterogeneities in turbid media: limits for detection, characterization, and comparison with absorption,” Appl. Opt. **37**, 6833–6844 (1998). [CrossRef]

19. S. Bjoern, S. V. Patwardhan, and J. P. Culver, “The influence of Heterogeneous optical properties upon fluorescence diffusion Tomography of small animals,” Springer Proc. in Physics **114**, 361–365 (2007). [CrossRef]

*a prior*knowledge of inhomogeneous optical properties improves the reconstructed image quality and quantification [17

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

19. S. Bjoern, S. V. Patwardhan, and J. P. Culver, “The influence of Heterogeneous optical properties upon fluorescence diffusion Tomography of small animals,” Springer Proc. in Physics **114**, 361–365 (2007). [CrossRef]

20. B. Brooksby, B.W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, J. Weaver, S.P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures using hybrid MRI-guided near-infrared spectral tomography,” Proceedings of the Natl. Acad. Sci. **103**, 8828–8833 (2006). [CrossRef]

21. Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. **10**, 024033–0240339 (2005). [CrossRef] [PubMed]

22. Q. Zhu, E. B. Cronin, A. A. Currier, H. S. Vine, M. Huang, N. Chen, and C. Xu, “Benign versus malignant breast masses: optical differentiation with US-guided optical imaging reconstruction,” Radiology **237**, 57–66 (2005). [CrossRef] [PubMed]

**30**, 861–869 (2003). [CrossRef] [PubMed]

9. A. Joshi, W. Bangerth, and E.M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

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

**12**, 5402–5417 (2004). [CrossRef] [PubMed]

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

## 2. Methodology

### 2.1 The model of diffusion equations

**12**, 5402–5417 (2004). [CrossRef] [PubMed]

**15**, 6955–6975 (2007). [CrossRef] [PubMed]

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

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

*denotes the photon density for excitation (subscript*

_{x,m}*x*) and fluorescence light (subscript

*m*). In the linear model employed in this paper

*µ*is the absorption coefficient and

_{ax,am}*D*=1/3 (

_{x,m}*µ*+µ

_{ax,am}*′*) is the diffusion coefficient of the tissue [3

_{sx,sm}3. M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**, 158–160 (1996). [CrossRef] [PubMed]

24. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express **13**, 9847–9857 (2005). [CrossRef] [PubMed]

*[3*

_{x}3. M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**, 158–160 (1996). [CrossRef] [PubMed]

*ηµ*(r) is the unknown fluorescence parameter to be reconstructed, where

_{af}*η*is the quantum yield and

*µ*is the absorption coefficient of fluorescence probe to excitation light.

_{af}6. N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. **32**, 382–384 (2007). [CrossRef] [PubMed]

7. H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. N. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt. **46**, 3617–3627 (2007). [CrossRef] [PubMed]

*r*(

_{sl}*l*=1,2,…,

*L*) represents the different excitation point source positions with respect to the subject with the amplitude Θ

*.*

_{s}25. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. **22**, 1779–1792 (1995). [CrossRef] [PubMed]

*n*⃗ denotes the outward normal vector to the surface and

*q*is a constant which is approximated

*q*≈(1+

*R*)/(1-

*R*) with

*R*=-1.4399

*k*

^{-2}+0.7099

*k*

^{-1}+0.6681+0.0636

*k*.

*k*is the ratio of optical reflective index of the inner tissue to that outside the boundary.

### 2.2 Finite element discretization

18. S. Srinivasan, B. W. Pogue, S. Davis, and F. Leblond, “Improved quantification of fluorescence in 3-D in a realistic mouse phantom,” Proc. SPIE **6434**, 64340S (2007). [CrossRef]

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

24. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express **13**, 9847–9857 (2005). [CrossRef] [PubMed]

18. S. Srinivasan, B. W. Pogue, S. Davis, and F. Leblond, “Improved quantification of fluorescence in 3-D in a realistic mouse phantom,” Proc. SPIE **6434**, 64340S (2007). [CrossRef]

_{x}(r) and Φ

*(*

_{m}*r*) in Eq. (1) can be equivalent obtained by solving the following weak form equations:

*ψ*(

*r*).

*N*vertex node and

_{p}*N*elements and employing

_{e}*ψ*(r) as the shape function, Φ

_{i}*(r) can be represented by*

_{x,m}*denotes the nodal value of Φ*

_{xi,mi}*(r)on the vertex*

_{x,m}*V*.

_{i}*ηµ*(r) be denoted as

_{af}*x*(r). Then it can be approximately expressed as

*γ*could be chosen to be the same as

_{i}*ψ*.

_{i}### 2.3 Generation of the linear scheme

*r*(

_{sl}*l*=1,2,…, L) with respect to the subject, the corresponding Φ

*is directly obtained by solving Eq. (6).*

_{x,sl}*is symmetrical and positive definite, Eq. (7) can be changed to*

_{m}*.*

_{x,sl}*} can be divided into two parts {Φ*

_{m,sl}

^{Meas}*} and {Φ*

_{m, sl}

^{NonM}*} where the former consists of the vertices on the surface for measurement and the latter is the remainder of the vertices. Removing the corresponding rows as {Φ*

_{m,sl}

^{NonM}*} from [*

_{m, sl}*B*], the following matrix equation is formed as follows:

_{sl}#### 2.4 The hybrid adaptive finite element (AFE) reconstruction algorithm

#### 2.4.1 Background

**30**, 861–869 (2003). [CrossRef] [PubMed]

**15**, 6955–6975 (2007). [CrossRef] [PubMed]

*a priori*information for the inversion problem while the others can inherit an initial value of the unknown parameter from a previous reconstruction procedure. Considering the above difference between the first reconstructed procedure and the succeeding ones, we proposed a hybrid adaptive finite element reconstruction algorithm.

#### 2.4.2 The hybrid AFE reconstruction algorithm

_{1},…Θ

*,…} in the AFE based reconstruction algorithm, a linear relationship as (10) could be generated as*

_{k}*is ill-conditioned and the boundary measurements contains noise. The common used Tikhonov regularization is always employed to find the regularized solution, as the minimizer of the following weighted combination of the residual norm and the side constraint:*

_{k}_{1}, where there is no

*a priori*knowledge about the distribution of fluorescence targets and all the vertices are included in the inversion. As an effective iterative regularization method, the CG algorithm is employed in the first reconstruction procedure. It is applied to the normal equation

*A*

^{T}

_{1}

*A*

_{1}

*X*

_{1}=

*A*

^{T}

_{1}Φ

*, where the low-frequency components of the solution tend to converge faster than the high-frequency components. Hence, it has some inherent regularization effect where the number of iterations plays the role of the regularization parameter [28]. The initial value of*

^{mea}*X*

_{1}is set to zero and the optimized iteration number is determined by L-curve criterion [29

29. P. C. Hansen, “Analysis of Discrete ill-posed problems by means of the L-curve,” SIAM Review **34**, 561–580 (1992). [CrossRef]

*a posteriori*error estimation of maximum selection, the elements with greater reconstructed value are selected to be refined [27]. In the implementation, the elements with the average value of the four vertices that is no less than 60% of the maximum value are selected for refinement each time. The boundary mesh of large value is also selected to be refined. Unlike other previous reports we use the longest refinement method to divide the tetrahedral element to second generation elements as showed in Fig. 2. That is, only the longest element edge is refined.

*X*

^{0}

_{k+1}of the (

*k*+1)

*th*(

*k*>=1) mesh level inherits the final solution

*X*of the

_{k}*k th*level by linear interpolation as follows:

*k*+1)

*th*mesh level is performed. In the proposed algorithm, the reconstructed result in the previous mesh level not only guides mesh refinement and provides an initial value for the refined mesh, but also determines a permissible region for the location of the fluorescence probes. In our study we choose nodes with the top 60% values in 0 k+1

*X*as the permissible region for the location of fluorescence targets. That is, the nodes values outside the permissible region are each set to zero. Thus, columns in

*A*+1(

*k*>=1)corresponding to the nodes outside the permissible region are removed. The matrix equation in (

*k*+1)

*th*(

*k*>=1)mesh level becomes

*A*=Φ

^{per}_{k+1}X^{per}_{k+1}

^{mea}_{k+1}.

*k*+1)

*th*(

*k*>=1) mesh level where

*a priori*knowledge of initial value and the permissible region are supplied by the previous reconstructed result, the following Landweber iterative regularization method is employed as [30

30. L. Landweber, “An iteration formula for Fredholm integaral equations of the first kind,” Am. J. Math. **73**, 615–624 (1951). [CrossRef]

^{X}

^{n+1}

*k*

^{+1}and

*Xnk*

_{+1}denote the iteration numbers of Landweber method and

*α*=1/

*λ*

_{max}with

*λ*

_{max}, the maximum eigenvalue of

*A*

^{per}k_{+1}·(

*A*

^{per}_{k+1})

*. The terminating iteration number of Landweber method is determined by both the discrepancy principle and the known value range of fluorescent yield [31*

^{T}31. G. A. Latham, “Best L^{2} Tikhonov Analogue for Landweber Iteration,” Inverse Probl. **14**, 1527–1537 (1998) [CrossRef]

## 3. Simulations and results

^{™}was employed to generate and solve the forward problem. Reconstructions were carried out on a personal computer with 2.8 GHz Pentium 4 processor and 1.5 GB RAM.

### 3.1 Experimental setup

6. N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. **32**, 382–384 (2007). [CrossRef] [PubMed]

7. H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. N. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt. **46**, 3617–3627 (2007). [CrossRef] [PubMed]

32. B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D Whole Body Mouse Atlas from CT and Cryosection Data,” Phys. Med. Biol. **52**, 577–587 (2007). [CrossRef] [PubMed]

*mm*was chosen as the region to be investigated. The rotational axis of the mouse was defined as the z axis with the bottom plane set as

*z*=0. Different views of the geometrical model are shown in Figs. 3(c) and 3(d). One unit in the model equals 0.1

*mm*. To simplify the problem, we considered the optical property outside the kidneys to be homogenous. Optical parameters were assigned as

*µ*=0.12

_{a}*mm*

^{-1}and µ

*′*=1.2

_{s}*mm*

^{-1}inside the kidneys (the blue part in Figs. 3(c) and 3(d)) and

*µ*=0.23

_{a}*mm*

^{-1}and µ

*′*=1.0

_{s}*mm*

^{-1}outside the kidneys (the green part in Figs. 3(c) and 3(d)) [34

34. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. **50**, 4225–4241 (2005). [CrossRef] [PubMed]

35. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**, 389–399 (2007). [CrossRef] [PubMed]

*′*) beneath the surface [25

_{s}25. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. **22**, 1779–1792 (1995). [CrossRef] [PubMed]

*z*=7.5

*mm*plane, as shown in Fig. 3(b). Therefore, two images were collected for each projection.

*mm*

^{3}.Then finite element method was applied to solve Eq. (1) and calculate the photon density distribution. Figure 4 shows an example of the geometry and the mesh when a fluorescence probe was placed in the left kidney. The geometry of the mouse abdomen part was discretized into 12832 nodes and 66047 tetrahedral elements. Figure 4(c) is the boundary view of the forward mesh and Fig. 4(d) is a different view for the internal part with

*z*<=8

*mm*. To simulate the real case, one percent of random Gaussian noise was then added to the calculated photon density distributions before applying the inverse algorithm.

### 3.2 Reconstruction for a single target

*mm*diameter and a 0.6

*mm*height was embedded in the left kidney. The fluorescent yield was set to 0.5. The reconstruction was carried out using the proposed algorithm. The maximum mesh level was set to

*K*

_{max}=3. The geometrical model in Fig. 3(c) and Fig. 3(d) was initially discretized into 2263 nodes and 10715 elements, as shown in Figs. 5(a) and 5(b). There was a total of 8606 nodes of measurement in the first mesh level for all the 30 different excitation source positions. Six conjugate gradient (CG) iterations were performed in the first mesh level. The top 30% of the calculated node values were kept as the reconstructed result.

^{st}mesh level, 173s in 2

^{nd}mesh level and 197s in 3

^{rd}mesh level.

*A*-Φ

_{k}X_{k}*‖ for the iterations in each mesh level. The residual error during the computation of the first mesh level descends fastest, when using the conjugate gradient algorithm. In the succeeding mesh levels the RE decreases slowly and smoothly, indicating that the step length between the two neighbor Landweber iterations becomes much smaller. In Fig. 5(c) the first RE value in the second mesh level and the third mesh level is much larger than the rest values in the same level because of the error brought by the linear interpolation from the reconstructed result of the previous level.*

^{meak}### 3.3 Reconstruction of double targets

*mm*in diameter and 0.6

*mm*in height with the fluorescent yield of 0.5. Both of the targets were embedded outside the kidney region with an edge-to-edge distance of 2

*mm*. Figure 8 shows the reconstruction results in each mesh level. In the third level the two targets could be resolved clearly although their positions are nearer to the surface than that of the real targets. The reconstruction result is also summarized in Table 2.

### 3.4 Comparison with a fixed mesh

## 5. Discussion and conclusion

*mm*. Two fluorescent targets with an edge-to-edge distance of 2

*mm*were also reconstructed. The results show that the two targets were resolved clearly. Compared with the regular finite element-based method using a fixed mesh, the proposed algorithm can reconstruct the shapes, the positions and the yield values more accurately. For reconstruction of double targets, our algorithm shows an improved spatial resolution compared with that of the regular method using a fixed mesh. Moreover, the reconstructed image quality and the spatial resolution could be further improved by optimizing a number of the system setup parameters such as the projection numbers, distribution of excitation sources and the detection range [35

35. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**, 389–399 (2007). [CrossRef] [PubMed]

**15**, 6955–6975 (2007). [CrossRef] [PubMed]

*a priori*knowledge. In this case, the conjugate gradient (CG) method converges very fast and is able to find an approximate value close to the real distribution of fluorescence yield in only a few iteration steps. Differently, the reconstructions in the succeeding mesh levels inherit an initial value from the previous reconstructed result. Based on

*a posteriori*error estimation of maximum selection, a small permissible region where the fluorescent targets are located can be determined. In this study, Landweber iteration algorithm was employed for the inversion of the mesh levels other than the initial level. It has a small step length between neighboring iterations, which could lead to improved reconstruction [36

36. W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. **10**, 1065–1069 (1999). [CrossRef]

*in-vivo*free-space fluorescence molecular imaging for small animals, two major issues exist: there is a need for reconstruction using the real animal-shape model and there is a need to take into consideration the heterogeneous optical properties of tissue. Several reports considered the importance of the heterogeneous nature of optical properties for the reconstruction for fluorescence tomography [17

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

19. S. Bjoern, S. V. Patwardhan, and J. P. Culver, “The influence of Heterogeneous optical properties upon fluorescence diffusion Tomography of small animals,” Springer Proc. in Physics **114**, 361–365 (2007). [CrossRef]

**114**, 361–365 (2007). [CrossRef]

16. 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]

12. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging **23**, 492–500 (2004). [CrossRef] [PubMed]

14. D. Hyde, A. Soubret, J. Dunham, T. Lasser, E. Miller, D. Brooks, and V. Ntziachristos, “Analysis of reconstructions in full view fluorescence molecular tomography,” Proc. SPIE **6498**, 649803 (2007). [CrossRef]

18. S. Srinivasan, B. W. Pogue, S. Davis, and F. Leblond, “Improved quantification of fluorescence in 3-D in a realistic mouse phantom,” Proc. SPIE **6434**, 64340S (2007). [CrossRef]

38. S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction,” J. Biomed. Opt. **10**, 050501-1:3(2005). [CrossRef]

**6434**, 64340S (2007). [CrossRef]

## Acknowledgments

## References and links

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

2. | V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. |

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

4. | J. Chang, H. L. Graber, and R. L. Barbour, “Luminescence optical tomography of dense scattering media,” JOSA A |

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

6. | N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. |

7. | H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. N. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt. |

8. | X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. |

9. | A. Joshi, W. Bangerth, and E.M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express |

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

11. | D. Wang, X. Song, and J. Bai, “A novel adaptive mesh based algorithm for fluorescence molecular tomography using analytical solution,” Opt. Express |

12. | R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging |

13. | R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett. |

14. | D. Hyde, A. Soubret, J. Dunham, T. Lasser, E. Miller, D. Brooks, and V. Ntziachristos, “Analysis of reconstructions in full view fluorescence molecular tomography,” Proc. SPIE |

15. | X. Li, B. Chance, and A. G. Yodh, “Fluorescent heterogeneities in turbid media: limits for detection, characterization, and comparison with absorption,” Appl. Opt. |

16. | 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 |

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

18. | S. Srinivasan, B. W. Pogue, S. Davis, and F. Leblond, “Improved quantification of fluorescence in 3-D in a realistic mouse phantom,” Proc. SPIE |

19. | S. Bjoern, S. V. Patwardhan, and J. P. Culver, “The influence of Heterogeneous optical properties upon fluorescence diffusion Tomography of small animals,” Springer Proc. in Physics |

20. | B. Brooksby, B.W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, J. Weaver, S.P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures using hybrid MRI-guided near-infrared spectral tomography,” Proceedings of the Natl. Acad. Sci. |

21. | Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. |

22. | Q. Zhu, E. B. Cronin, A. A. Currier, H. S. Vine, M. Huang, N. Chen, and C. Xu, “Benign versus malignant breast masses: optical differentiation with US-guided optical imaging reconstruction,” Radiology |

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

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

25. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. |

26. | S. S. Rao, “The finite element method in engineering,” (Butterworth-Heinemann, Boston, 1999). |

27. | W. Bangerth, “Adaptive finite element methods for the identification of distributed parameters in partial differential equations,” Ph.D. thesis, University of Heidelberg (2002). |

28. | M. Hanke and P. C. Hansen, “Regularization methods for large-scale problems,” Surv. Math. Ind. |

29. | P. C. Hansen, “Analysis of Discrete ill-posed problems by means of the L-curve,” SIAM Review |

30. | L. Landweber, “An iteration formula for Fredholm integaral equations of the first kind,” Am. J. Math. |

31. | G. A. Latham, “Best L |

32. | B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D Whole Body Mouse Atlas from CT and Cryosection Data,” Phys. Med. Biol. |

33. | D. Stout, P. Chow, R. Silverman, R. M. Leahy, X. Lewis, S. Gambhir, and A. Chatziioannou, “Creating a whole body digital mouse atlas with PET, CT and cryosection images,” Molecular Imaging and Biology. |

34. | G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. |

35. | T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. |

36. | W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. |

37. | L. H. Peng, G. Lu, and W. Q. Yang, “Image reconstruction algorithms for electrical capacitance tomography: state of the art,” J. Tsinghua Univ. (Sci & Tech) |

38. | S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction,” J. Biomed. Opt. |

**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

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

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: October 24, 2007

Revised Manuscript: December 3, 2007

Manuscript Accepted: December 17, 2007

Published: December 20, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Xiaolei Song, Daifa Wang, Nanguang Chen, Jing Bai, and Hongkai Wang, "Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm," Opt. Express **15**, 18300-18317 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-18300

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### References

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- S. Srinivasan, B. W. Pogue, S. Davis, and F. Leblond, "Improved quantification of fluorescence in 3-D in a realistic mouse phantom," Proc. SPIE 6434, 64340S (2007). [CrossRef]
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- B. Brooksby, B.W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, J. Weaver, S.P. Poplack, and K. D. Paulsen, "Imaging breast adipose and fibroglandular tissue molecular signatures using hybrid MRI-guided near-infrared spectral tomography," Proc. Natl. Acad. Sci. 103, 8828-8833 (2006). [CrossRef]
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- S. S. Rao, The Finite Element Method in Engineering, (Butterworth-Heinemann, Boston, 1999).
- W. Bangerth, "Adaptive finite element methods for the identification of distributed parameters in partial differential equations," Ph.D. thesis, University of Heidelberg (2002).
- M. Hanke and P. C. Hansen, "Regularization methods for large-scale problems," Surv. Math. Ind. 3, 253-315 (1993).
- P. C. Hansen, "Analysis of Discrete ill-posed problems by means of the L-curve," SIAM Rev. 34, 561-580 (1992). [CrossRef]
- L. Landweber, "An iteration formula for Fredholm integaral equations of the first kind," Am. J. Math. 73, 615-624 (1951). [CrossRef]
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- D. Stout, P. Chow, R. Silverman, R. M. Leahy, X. Lewis, S. Gambhir, and A. Chatziioannou, "Creating a whole body digital mouse atlas with PET, CT and cryosection images," Mol. Imaging Biol. 4, S27 (2002).
- G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, "Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study," Phys. Med. Biol. 50, 4225-4241 (2005). [CrossRef] [PubMed]
- T. Lasser and V. Ntziachristos, "Optimization of 360o projection fluorescence molecular tomography," Med. Image Anal. 11, 389-399 (2007). [CrossRef] [PubMed]
- W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, "An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography," Meas. Sci. Technol. 10, 1065-1069 (1999). [CrossRef]
- L. H. Peng, G. Lu, and W. Q. Yang, "Image reconstruction algorithms for electrical capacitance tomography: state of the art," J. Tsinghua Univ. Meas. Sci. Technol. 44, 478-484 (2004).
- S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, "Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction," J. Biomed. Opt. 10, 050501-1:3 (2005). [CrossRef]

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