Reconstruction for free-space fluorescence ... - OSA Publishing

Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm Xiaolei Song1, Daifa Wang1, Nanguang Chen2,3, Jing Bai1*, and Hongkai Wang1 1 Department of Biomedical Engineering, Tsinghua University, Beijing 100084, China Division of Bioengineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore 3 Department of Electrical & Computer Engineering, National University of Singapore, 9 Engineering Dr. 1, Singapore 117576, Singapore *Corresponding author: [email protected]

2

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 360o 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 OCIS codes: (170.3010) Image reconstruction Techniques; (170.6280) Spectroscopy, fluorescence and luminescence; (170.5280) Photon Migration; (170.3880) Medical and Biological Imaging

References and links 1.

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Received 24 Oct 2007; revised 3 Dec 2007; accepted 17 Dec 2007; published 20 Dec 2007

24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 18300

13. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett. 28, 1701-1703 (2003). 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). 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). 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). 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). 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). 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). 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). 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). 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). 23. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite element based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998). 24. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847-9857 (2005). 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). 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. 3, 253-315 (1993). 29. P. C. Hansen, “Analysis of Discrete ill-posed problems by means of the L-curve,” SIAM Review 34, 561580 (1992). 30. L. Landweber, “An iteration formula for Fredholm integaral equations of the first kind,” Am. J. Math. 73, 615-624 (1951). 31. G. A. Latham, “Best L2 Tikhonov Analogue for Landweber Iteration,” Inverse Probl. 14, 1527-1537 (1998) 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). 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. 4, S27 (2002). 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). 35. T. Lasser and V. Ntziachristos, “Optimization of 360o projection fluorescence molecular tomography,” Med. Image Anal. 11, 389-399 (2007). 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). 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) 44, 478-484 (2004). 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).

1. Introduction In recent years, fluorescence molecular tomography (FMT) has emerged as a promising tool for 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

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made to develop new probes, photon migration models, imaging systems, and the corresponding reconstruction methods [1, 2]. During the last decade, imaging systems for fluorescence tomography have evolved over several generations from the early fiber-based systems [3, 4] to the non-contact detection system of slab geometry using charge coupled device (CCD) [5]. Owing to the large number of spatial samples, the latter leads to a sub-millimeter resolution. However, both the two kinds of systems require matching fluid, which is inconvenient for experiments. In 2007 a freespace FMT with full 360o geometry projections was reported [6], which could capture the 3-D surface shape of the imaging animal and get rid of the matching fluid [7]. Additionally, it overcame the limited projection angles of the slab geometry while keeping the high sampling of photon field. Ultimately, this imaging approach is fairly flexible to integrate with other imaging modalities such as X-ray CT or MRI, supplying anatomic information as well. With the improvement of physical systems for photon collection, there is a need for researches on new reconstruction methods for FMT [1, 2]. On the one hand, the high sampling measurements and the real animal-shape geometry lead to large datasets, which require long computational time and big storage capacity. So the computational efficiency of the reconstruction method need to be improved [2]. On the other hand, an inversion method, which is appropriate for one system, may not function properly for different systems. In the past four years adaptive finite element (AFE) and adaptive meshing techniques have been employed for reconstruction in optical tomography. In 2003 Gu, et al. [8] proposed a reconstruction method using adaptive meshing for 2-D diffuse optical tomography. The results using phantom measurements demonstrated both qualitative and quantitative improvement of optical image reconstruction. For fluorescence tomography, Joshi et al. [9] developed in 2003 an AFE method for reconstructing fluorescent targets in a reflectance cube geometry using hexahedral mesh. Recently Lee and Joshi et al. [10] presented a fully adaptive finite element based reconstruction algorithm using tetrahedral elements, which was applied to the reconstruction of human breast geometry using point source illumination and detection. In addition to the above works based on finite element analysis, Wang and Song et al. [11] applied adaptive meshing technique in FMT of slab geometry, using the analytical solution of diffusion equations. Compared with the globally uniform discretization, adaptive methods have been shown to significantly reduce the data size and improve the computational efficiency. Additionally they could stabilize the solution of the inversion problem while providing the necessary resolution for FMT of small animal [9-11]. The adaptive reconstruction methods were realized by locally refining the particular region based on 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 [8-11] use the same inversion techniques for all their different discretization levels, without considering the differences between the first discretization level and the succeeding ones. Furthermore, the previous adaptive reconstruction algorithms [8-11] were demonstrated using regular geometry such as cylinder, hemisphere, cube, or a combination of these geometries with spatially homogenous optical parameters. However, for free-space imaging of small animals, the reconstruction should be based on an irregular shape of the animal being imaged, which leads to a more accurate description of the forward model and reconstruction results than the regular shape [12-14]. Additionally, the influence of tissue optical heterogeneity on fluorescence tomography reconstruction has been studied for years [15-19]. It has been proved that a prior knowledge of inhomogeneous optical properties improves the reconstructed image quality and quantification [17-19]. The distributions of optical parameters corresponding to different organs could be obtained by diffusion optical tomography, utilizing the anatomic information from co-registered MRI [20], X-ray CT [21] and ultrasound [22]. Dual meshing strategy was employed in [8], [9] and [10], where both the forward and inverse meshes were adaptively refined. In fluorescence tomography the inversion was treated as a nonlinear optimization problem and solved it by a truncated Gauss-Newton (GN) strategy #88964 - $15.00 USD

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[9] or GN with Steihaug-Conjugate Gradient [10]. However, it required that the forward model with updated parameters had to be computed at each iteration procedure. So the computational cost and programming difficulty was increased especially when using the adaptive algorithm. In this paper we report the development of a novel hybrid adaptive finite element algorithm for reconstruction of FMT, 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, considering their different characteristics. The new algorithm was validated using surface data generated by numerical simulations, based on the latest free-space system of a rotating subject. The algorithm was applied to a 3-D digital mouse model, incorporating the known optical heterogeneity between different tissues. The results suggest that we are able to achieve high computation efficiency and spatial resolution for models with irregular shape and inhomogeneous distributions of optical parameters. This paper is organized as follows. In section 2, the diffusion approximation of photon propagation and its finite element solution are initially introduced. Then the formulation of the linear scheme and the novel hybrid adaptive finite element algorithm are presented. In section 3, numerical simulations with fluorescence targets embedded in a 3-D model from a mouse atlas are shown. Finally, we discuss the results and draw a number of conclusions in section 4. 2. Methodology 2.1 The model of diffusion equations In the near infrared spectral window, diffusion equation is always used to describe the photon migration in biological tissues. For the continuous wave FMT with point excitation sources, the model for photon propagation is usually presented using the following coupled diffusion equations [9, 10, 17, 23]:

⎧∇Dx (r)∇Φ x (r ) − μ ax (r )Φ x (r ) = −Θsδ (r − rsl ) ⎨ ⎩∇Dm (r )∇Φ m ( r ) − μam (r )Φ m (r ) = −Φ x ( r )ημ af (r )

(1.1)

,

(1)

(1.2)

where Φ x,m denotes the photon density for excitation (subscript x ) and fluorescence light (subscript m ). In the linear model employed in this paper μax, am is the absorption coefficient and Dx,m = 1 / 3(μax,am + μ 'sx,sm ) is the diffusion coefficient of the tissue [3, 24]. The absorption and scattering of the excitation light, caused by the fluorescence probes, were assumed to have little effect on the distribution of Φ x [3], as the fluorescence probes often occupy a very small volume compared to the total imaging area. The fluorescent yield ημ af (r) is the unknown

η

is the quantum yield and μ af is the absorption coefficient of fluorescence probe to excitation light. Whereas the FMT with 360o projections is carried out by rotating the subject while the excitation source and the detection device are stationary [6, 7], for the reconstruction we consider the phantom or animal geometry model to be stationary while the source and the corresponding detected boundary part to be moving. On the right hand side of Eq. (1.1) rsl (l = 1,2,..., L) represents the different excitation point source positions with respect to the

fluorescence parameter to be reconstructed, where

subject with the amplitude Θs . To solve these equations, the Robin-type boundary conditions are implemented on the boundary ∂Ω of domain Ω [25]: (2) 2 Dx ,m ∂Φ x ,m / ∂n + qΦ x ,m = 0, where n denotes the outward normal vector to the surface and q is a constant which is approximated

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as 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 The finite element method has been widely used for numerically solving the diffusion equations [18, 23, 24], especially with arbitrary geometries [18]. Based on the finite element theory [26] , Φ x ( r ) and Φ m (r ) in Eq. (1) can be equivalent obtained by solving the following weak form equations: 1 ⎧ ⎪ ∫ Ω (D x ( r )∇ Φ x ( r ) ⋅ ∇ ψ ( r ) + μ ax ( r ) Φ x ( r )ψ ( r ))dr + ∫ ∂Ω 2 q Φ x ( r )ψ ( r ) dr ⎪ ⎪ (3.1) = ∫ Ω Θ sδ ( r − rsl )ψ (r)dr ⎪ , ⎨ 1 ⎪ (D ( r )∇ Φ ( r ) ⋅ ∇ ψ ( r ) + μ ( r )Φ ( r )ψ ( r )) dr + Φ m ( r )ψ ( r )dr m am m ∫ ∂Ω ⎪∫ Ω m 2q ⎪ (3.2) = ∫ Ω Φ x ( r )ημ af ( r )ψ (r)dr ⎪ ⎩

(3)

by implementing the test function ψ (r ) . Discretizing the domain Ω with N p vertex node and N e elements and employing ψ i (r ) as the shape function, Φ x ,m (r ) can be represented by

Φ

x ,m

(r) =

∑

N

p

i =1

Φ

xi , mi

ψ i ( r ),

(4)

where Φ xi ,mi denotes the nodal value of Φ x ,m (r ) on the vertex Vi .

Let { γ 1 , γ 2 ,⋅ ⋅ ⋅, γ N p } be the basis functions and the unknown fluorescent yield ημ af ( r ) be denoted as x ( r ) . Then it can be approximately expressed as

x(r ) = ημ af (r ) =

∑

Np

i =1

(ημ af )i γ i (r ) =

∑

x ⋅ γ i (r ),

Np i =1 i

(5)

where γ i could be chosen to be the same as ψ i . By incorporating Eq. (4) and Eq. (5) into Eq. (3), the following matrix equations are obtained

[ K x ]{Φ x } = {Lx }

(6)

[ K m ]{Φ m } = [ F ]{ X }

(7)

1 ⎧ with ⎪⎨ K xi , xj = ∫ Ω (D x ∇ψ x i ⋅ ∇ψ xj + μ axψ x iψ x j ) dr + 2 q ∫ ∂Ωψ x iψ x j dr and ⎪ Lxi , xj = ∫ Ω Θ sδ ( r − rs )ψ xj dr ⎩ 1 ⎧ ⎪ K mi, mj = ∫ Ω (D m ∇ψ m i ⋅ ∇ψ mj + μ amψ m iψ m j ) dr + ∂Ωψ m iψ m j dr . 2q ∫ ⎨ ⎪ f i , j = ∫ Ω Φ x (r )ψ miψ mj dr ⎩

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2.3 Generation of the linear scheme

Assuming that the optical properties of both the excitation and the emission are known, a linear relationship between the measured fluorescence data on the boundary and the unknown internal fluorescent yield can be generated by the following steps: Step1. For each excited source position at rsl (l = 1,2,..., L) with respect to the subject, the corresponding Φ x,sl is directly obtained by solving Eq. (6). Step2. As the stiff matrix K m is symmetrical and positive definite, Eq. (7) can be changed to

{Φ m ,sl } = [ K m−1 ] ⋅ [ Fsl ] ⋅ { X } = [ Bsl ] ⋅ { X },

(8)

for a specific excitation light distribution Φ x, sl . Step3. {Φ m ,sl } can be divided into two parts {Φ Meas and {Φ mNonM where the former m, sl } ,sl } consists of the vertices on the surface for measurement and the latter is the remainder of the vertices. Removing the corresponding rows as {Φ mNonM from [ Bsl ] , the following matrix , sl }

equation is formed as follows

：

{Φ Meas m, sl } = [ Asl ] ⋅{ X }.

(9)

Step4. Assembling Eq. (9) for different excitation cases, the final weighted matrix for a particular reconstruction mesh system is generated as:

⎧Φ Meas ⎫ ⎡ A ⎤ ⎧ημ af (1) ⎫ m , s1 ⎪ Meas ⎪ ⎢ s1 ⎥ ⎪ ⎪ ⎪Φ m,s 2 ⎪ ⎢ As 2 ⎥ ⎪ημ af ( 2 ) ⎪ ⎨ ⎬ = ⎢ ⎥⎨ ⎬ = [ A]{ X }. ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪Φ Meas ⎪ ⎣ AsL ⎦ ⎪ημ af ( Np ) ⎪ ⎭ ⎩ ⎩ m,sL ⎭

(10)

2.4 The hybrid adaptive finite element (AFE) reconstruction algorithm 2.4.1 Background In adaptive finite-element based reconstruction technique [27], the imaging domain Ω is not divided into a fixed, uniformly fine mesh so as to achieve the necessary resolution. Instead, Ω is dynamically discretized in several levels from a coarser mesh to a finer mesh with local refinement of the region of interest. A typical AFE based reconstruction algorithm [8-10] includes the following three steps: (1) The unknown parameter is reconstructed firstly utilizing a uniformly coarse mesh, where a rough estimation of the parameter is obtained; (2) Based on the previous reconstructed result, a particular portion of the imaging domain is selected and refined, which lead to a new mesh with the inherited initial value of the required parameter; (3) A new reconstruction procedure is carried out on the refined mesh. If the quality of the reconstructed image is not satisfied, a new iteration of mesh refinement and reconstruction will be repeated. It is obvious that the first reconstructed procedure is different from the succeeding ones in two aspects: (1) It is based on a uniformly mesh while others are with a locally refined mesh. (2) It has no 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.

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2.4.2 The hybrid AFE reconstruction algorithm Supposing there is a sequence of tetrahedral mesh levels {Θ1 ,...Θ k ,...} in the AFE based reconstruction algorithm, a linear relationship as (10) could be generated as

Ak X k = Φ mea k

(11)

for each mesh level. In fluorescence tomography these are usually ill-posed problems; That is, the weight matrix A 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:

min

x ≤ xk ≤ x L

E k ( X k ) = A k X k − Φ mea k h

2 2

+ λ kη k ( X k ).

(12)

Usually iterative regularization methods are used for image reconstruction. In the hybrid adaptive finite element algorithm, two different iterative inversion techniques are separately applied to the first mesh level and the succeeding levels. The procedure of the proposed algorithm is illustrated in Fig. 1. Firstly the imaging domain is discretized into a uniformly coarse mesh Θ1 , 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 A1T A1 X 1 = A1T Φ1mea , 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 X 1 is set to zero and the optimized iteration number is determined by L-curve criterion [29] or by experience. The CG method converges very fast and is able to find an approximate value close to the real distribution in only a few iteration steps. After the iteration in the first level stops, mesh refinement is applied. According to 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. When switching to a finer mesh, the initial value X k0+1 of the (k + 1)th ( k >= 1) mesh level inherits the final solution X k of the k th level by linear interpolation as follows:

X k0+1 = Fkk +1 ( X k )

(13) Then, the optimization in (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 X k0+1 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 Ak +1 (k >= 1) corresponding to the nodes outside the permissible region are removed. The matrix equation per mea in (k + 1)th (k >= 1) mesh level becomes Akper . +1 X k +1 = Φ k +1

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Begin

Form the geometry model and discretize it with 1st finite element mesh level Θ1

Establish the linear equation of A1 X 1 = Φ1mea in Θ1

N 1 step of CG iteration

Refine selected elements according to maximum selection rule, obtain (k + 1 )th ( k >= 1 ) mesh level Θ k

Take nodes within top 60% value as permissible per mea region, obtain matrix equation Akper +1 X k +1 = Φ k +1

N k +1 steps of Landweber iteration

Yes

Φ mea − Φ cal > ε or k < K max

No end

Fig. 1. Flow diagram for the hybrid adaptive finite element reconstruction algorithm

(a)

(b)

Fig. 2. Mesh refinement method. (a) is the original tetrahedral element. (b) is the second generation elements after refinement.

For the (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] T mea per n X kn++11 = X kn+1 + α ( Akper +1 ) (Φ k +1 − Ak +1 X k +1 ),

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(14)



where the superscripts of X kn++11 and X kn+1 denote the iteration numbers of Landweber method per T and α = 1 / λmax with λmax , the maximum eigenvalue of Akper .The terminating +1 ⋅ ( Ak +1 ) iteration number of Landweber method is determined by both the discrepancy principle and the known value range of fluorescent yield [31]. Usually Landweber method converges slowly. However, as a permissible region for the location of fluorescence targets is determined in the mesh levels other than the first one and the size of the corresponding weighted matrix is reduced, the convergence process is accelerated in this situation. Additionally, the reconstruction value is also stabilized, due to the small step length between neighboring iterations.

3. Simulations and results

A series of computerized simulations were designed to test the new algorithm. Most parts of the algorithm were coded in Matlab for flexibility. A commercial finite element software COMSOL MultiphysicsTM 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.

(a)

(b)

(c)

(d)

Fig. 3. Experimental setup and the geometrical model for a mouse abdomen part. The sketch of the free-space FMT system is illustrated in (a), which is similar to that in Ref. [6]. The plane of excitation sources is shown in (b) where the black points represent positions of the isotropic point sources. The points 01-015 are the first excitation sources for the 15 projections and the points 016-030 are the second ones. The field of view (FOV) of 150o for detection with respect to the excitation source point 01 is also shown. (c) and (d) are different views of the mouse geometry model used in the reconstruction. In order to reduce the boundary artifacts, which interfere with the finite element computation, the model was generated by sampling the original atlas data (intersections of the vertical and horizontal lines) and then approximating the curves using spline function to form the kidney surface and the body surface.

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3.1 Experimental setup

The synthetic measurements were generated based on an imaging configuration of the freespace FMT with 360o geometry projections [6, 7]. The sketch of the system structure is shown in Fig. 3(a). The mouse being imaged is suspended on a rotating stage, with a laser beam focused on its surface. A highly sensitive CCD camera is placed on the opposite side of the laser, collecting the fluorescent photon fields propagating through the mouse. A 3-D mouse atlas of CT and cryosection data was employed to provide 3-D anatomical information [32, 33] (http://neuroimage.usc.edu/Digimouse.html). Firstly the mouse atlas was imported to COMSOL MultiphysicsTM to form the 3-D geometry model. In our simulations, the abdomen part of the mouse atlas with a height of 15mm 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 μa = 0.12 mm−1 and μ s ' = 1.2 mm −1 inside the kidneys (the blue part in Figs. 3(c) and 3(d)) and μa = 0.23 mm−1 and μ s ' = 1.0 mm −1 outside the kidneys (the green part in Figs. 3(c) and 3(d)) [34]. In this simulation fluorescence tomography of 360o full view was performed using 15 projections with 150 field of view detected for each projection [35]. The collimated laser beam was modeled as an isotropic point source, located one mean free path of photon transport ( 1 / μ s ' ) beneath the surface [25]. For each projection there were 2 point source positions of 18o interval at the z = 7.5 mm plane, as shown in Fig. 3(b). Therefore, two images were collected for each projection.

(a)

(b)

(c)

(d)

Fig. 4. The forward geometry model for generating surface measurements is shown in (a) and (b) with a fluorescent probe in the left kidney, where the red part represents the fluorescent probe. Different views for the mesh of tetrahedral elements are shown in (c) and (d).

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(C) 2007 OSA



To generate the synthetic measurements on the surface, the mouse geometry model was discretized into a fine tetrahedral-element mesh with the maximum element size of 0.57mm3 .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